Strict Convexity of the Surface Tension for Non-convex Potentials
aa r X i v : . [ m a t h - ph ] J un Strict Convexity of the Surface Tensionfor Non-convex Potentials
Stefan AdamsRoman Koteck´yStefan M¨uller
Author address:
Mathematics Institute, University of Warwick, Coventry CV4 7AL,United Kingdom
E-mail address : [email protected]
Mathematics Institute, University of Warwick, Coventry CV4 7AL,United Kingdom and Center for Theoretical Study, Charles Univer-sity, Jilsk´a 1, Prague, Czech Republic
E-mail address : [email protected]
Universit¨at Bonn, Endenicher Allee 60, D-53115 Bonn, Germany
E-mail address : [email protected] ontents Acknowledgment 1Chapter 1. Introduction 3Chapter 2. Setting and Results 72.1. Setup 72.2. Main result 82.3. Proofs of the given examples 12Chapter 3. The Strategy of the Proof 19Chapter 4. Detailed Setting of the Main Steps 234.1. Finite range decomposition. 234.2. Polymers, polymer functionals, ideal Hamiltonians and norms. 254.3. Definition of the renormalisation transformation T k : ( H k , K k ) ( H k +1 , K k +1 ) 294.4. Key properties of the renormalisation transformation 334.5. Fine tuning of the initial conditions 374.6. Proof of strict convexity—Theorem 2.1 39Chapter 5. Properties of the Norms 43Chapter 6. Smoothness 536.1. Immersion E : M → M |k P P R R P C ( q ) A ( q ) − and B ( q ) F H iiiv CONTENTS a) d = 1 103b) Multidimensional case 104Appendix C. Gaussian Calculus 107Appendix D. Chain Rules 115D.1. Motivation 115D.2. Derivatives and their relations 116D.3. Chain rule with a loss of regularity 123D.4. Chain rule with parameter and a graded loss of regularity 124D.5. A special case of a function G that is linear in its first argument 128D.6. A special case of function G not depending on the parameter p C \ C ∗ and failure of the inverse functions theorem in C ∗ bstract We study gradient models on the lattice Z d with non-convex interactions.These Gibbs fields (lattice models with continuous spin) emerge in various branchesof physics and mathematics. In quantum field theory they appear as massless fieldtheories. Even though our motivation stems from considering vector valued fieldsas displacements for atoms of crystal structures and the study of the Cauchy-Bornrule for these models, our attention here is mostly devoted to interfaces, with thegradient field as an effective interface interaction. In this case we prove the strictconvexity of the surface tension (interface free energy) for low temperatures andsufficiently small interface tilts using muli-scale (renormalisation group analysis)techniques following the approach of Brydges and coworkers [ Bry09 ]. This is acomplement to the study of the high temperature regime in [
CDM09 ] and it is anextension of Funaki and Spohn’s result [
FS97 ] valid for strictly convex interactions.
Mathematics Subject Classification.
Primary 82B28; Secondary 82B41; 60K60; 60K35.
Key words and phrases.
Renormalisation group; random field of gradients; surface tension;multi-scale analysis; loss of regularity. v cknowledgment We are grateful to David Brydges for generously sharing his ideas on renor-malisation group methods with us and for many interesting discussions. We thankDavid Preiss for inspiring discussion on differentiability properties and for provid-ing notes on which Appendix D is based. We also thank S. Buchholz, S. Hilger,G. Menz, F. Otto, E. Runa for many helpful suggestions and comments. The re-search of S. Adams was supported by EPSRC grant number EP/I003746/1 andby the Royal Society Exchange grant IE130438
The Challenge of Different Scalesin Nature . S. Adams thanks the mathematics department at UBC for the warmhospitality during his sabbatical stay 2013-2014. R. Koteck´y was supported bythe grants GA ˇCR P201/12/2613
Threshold phenomena in stochastic systems andGA ˇCR 16-15238S
Collective behavior of large stochastic systems and S. M¨uller bythe DFG Research group FOR 718
Analysis and stochastics in complex physicalsystems (20062013), by the Hausdorff Center for Mathematics (since 2008) and bythe CRC 1060
The mathematics of emergent effects (since 2013). HAPTER 1
Introduction
This paper has two related goals.First, we seek to identify uniform convexity properties for a class of latticegradient models with non-convex microscopic interactions.Secondly, we extend the rigorous renormalisation group techniques developedby Brydges and coworkes to models without a discrete rotational symmetry of theinteraction. In the presence of symmetry, the set of relevant terms is stronglyrestricted by the symmetry.Regarding the first goal, we consider gradient random fields { ϕ ( x ) } x ∈ L indexedby a lattice L with values in R m , ϕ ( x ) ∈ R m . The term gradient is referring to theassumption that the distribution depends only on gradients ∇ e ϕ ( x ) = ϕ ( x + e ) − ϕ ( x ).These type of fields are used as effective models of crystal deformation or phaseseparation. In the former case, where m = 3 and L ⊂ Z , the value ϕ ( x ) plays therole of a displacement of an atom labelled by a site x of a crystal under deformation.Even though the former case is our main motivation, we will restrict our attentionhere, for simplicity, to the latter case with m = 1 and L = Z d . This is a modeldescribing a phase separation in R d +1 with ϕ ( x ) ∈ R corresponding to the positionof the (microscopic) phase separation surface. The model is a reasonably effectiveapproximate description in spite of the fact that it ignores overhangs of separationsurface as well as any correlations inside and between the coexisting phases.The distribution of the interface is given in terms of a Gibbs distribution withnearest neighbour interactions of gradient type, that is, the interaction betweenneighboring sites x, x + e i depends only on the gradient ∇ i ϕ ( x ) = ϕ ( x + e i ) − ϕ ( x ) , i = 1 , . . . , d . More precisely, for any finite Λ ⊂ Z d we consider the Hamiltonianof the form H Λ ( ϕ ) = X x ∈ Λ d X i =1 W ( ∇ i ϕ ( x )) , where W : R → R is a perturbation of a quadratic functions, i.e. W ( η ) = 12 η + V ( η ) with some perturbation V : R → R . For a given boundary condition ψ ∈ R ∂ Λ , where ∂ Λ = { z ∈ Z d \ Λ : | z − x | =1 for some x ∈ Λ } , we consider the Gibbs distribution at inverse temperature β > γ ψ Λ ,β (d ϕ ) = 1 Z Λ ( β, ψ ) exp (cid:0) − βH Λ ( ϕ ) (cid:1) Y x ∈ Λ d ϕ ( x ) Y x ∈ ∂ Λ δ ψ ( x ) (d ϕ ( x )) , where the normalisation constant Z Λ ( β, ψ ) is the integral of the density and is calledthe partition function. One is particularly interested in tilted boundary conditions ψ u ( x ) = h x, u i , for some tilt u ∈ R d . An object of basic relevance in this context is the surface tension or free energy defined by the limit(1.1) σ β ( u ) = − lim Λ ↑ Z d β | Λ | log Z Λ ( β, ψ u ) . The surface tension σ β ( u ) can also be seen as the price to pay for tilting a macro-scopically flat interface. The existence of the above limit follows from a standardsub-additivity argument.In the case of a strictly convex potential, Funaki and Spohn show in [ FS97 ]that σ β is convex as a function of the tilt. The simplest strictly convex potentialis the quadratic one with V = 0, which corresponds to a Gaussian model, alsocalled the gradient free field. The convexity of the surface tension plays a crucialrole in the derivation of the hydrodynamical limit of the Landau-Ginsburg modelin [ FS97 ]. Strict convexity of the surface tension for strictly convex W with 0 FS97 ] and [ DGI00 ] use explicitly the conditions on thesecond derivative of W in their proof. In particular they rely on the Brascamp-Liebinequality and on the random walk representation of Helffer and Sj¨ostrand, whichrequires a strictly convex potential W .In [ CDM09 ] Deuschel et al showed the strict convexity of the surface tensionfor non-convex potentials in the small β (high temperature) regime for potentialsof the form W ( t ) = W ( t ) + g ( t ) , where W is strictly convex as above and where g ∈ C ( R ) has a negative boundedsecond derivative such that √ β k g ′′ k L ( R ) is sufficiently small. These studies havebeen applied in [ CD12 ] to large deviations principle for the profile.In the present paper, we show the strict convexity of the surface tension forlarge enough β (low temperatures) and sufficiently small tilt, using multi-scaletechniques based on a finite range decomposition of the underlying backgroundGaussian measure in [ AKM13 ].Note also that, due to the gradient interaction, the Hamiltonian has a contin-uous symmetry. In particular this implies that no Gibbs measures on Z d exist fordimensions d = 1 , FP81 ]. If one considers thecorresponding random field of gradients (discrete gradient image of the height field ϕ ) it is clear that its distribution depends on the gradient of the boundary condi-tion of the height field. One can also introduce gradient Gibbs measures in terms ofconditional distributions satisfying DLR equations, cf. [ FS97 ]. For strictly convexinteraction W with bounds on the second derivative, Funaki and Spohn in [ FS97 ]proved the existence and uniqueness of an extremal, i.e. ergodic, gradient Gibbsmeasure for each tilt u ∈ R d . In the case of non-convex W , uniqueness of theergodic gradient component can be violated, for tilt u = 0 this has been proved in[ BK07 ]. However in this phase transition situation in [ BK07 ], the surface tensionis not strictly convex at tilt u = 0. . INTRODUCTION 5 The second goal of the present paper is to show in detail how the rigorousrenormalisation approach of Brydges and coworkers (see [ BY90 ] for early work,[ Bry09 ] for a survey and [ BS15a, BS15b, BBS15a, BS15c, BS15d, BBS15b ]for recent developments which go well beyond the gradient models discussed inthis paper) can be extended to accommodate a class of models without a discreterotational symmetry of the interaction.In accordance with the general renormalization group strategy, the resultingpartition function Z Λ ( β, ψ u ) is obtained by a sequence of “partial integrations”(labelled by an index k ). The result of each of them is expressed in terms of twofunctions: the “irrelevant” polymers K k that are decreasing with each subsequentintegration, and the “relevant” ideal Hamiltonians H k —homogeneous quadraticfunctions of gradients ∇ ϕ parametrized by a fixed finite number of parameters. Tofine-tune the procedure so that the final integration yields a result with a straight-forward bound we need to assure the smoothness of the procedure with respectto the parameters of a suitably chosen “seed Hamiltonian”. However, it turns outthat the derivatives with respect to those parameters lead to a loss of regularity offunctions K k and H k considered as elements in a scale of Banach spaces.A more detailed summary of the strategy is presented in Chapter 3 wherethe reader can get an overview of our methods and techniques of the proof. First,however, we will summarize the main claims concerning the convexity of the surfacetension σ β ( u ) in Chapter 2. The detailed formulations and proofs are in Chapters 4–8. Miscelaneous technical details are deferred to Appendices.Various extensions and generalisations of our work are possible.First, Buchholz has very recently developed a new finite range decompositionfor which no loss of regularity occurs in the problem we study [ Buc16 ]. However,in the present paper we decided to stick to the usual finite range decompositionand to explain how the loss for regularity can be overcome by a suitable version ofthe chain rule and the implicit function theorem since we believe that these toolsmight be useful in other contexts, too.Secondly, we restrict ourselves to dimensions d = 2 and d = 3 because in thatcase there are only two types of linear relevant terms: linear combinations of thefirst and second discrete derivatives of the field. Our approach can be extended tohigher dimensions by including linear terms in higher derivatives of the field. Thisonly requires an extension of the appropriate “homogenisation projection operator”Π used in the definition of quadratic functions H k (see Chapter 4.3) to relevantpolynomials and the corresponding discrete Poincar´e type inequalities. In fact,Brydges and Slade [ BS15b ] have recently developed a very general theory whichallows one to define the projection onto the relevant polynomials and to prove thenecessary estimates.Thirdly, we focus on scalar valued field even though most our methods carrydirectly over to the vector valued case which is relevant in elasticity. The discussionof models relevant in elasticity requires, however, also a number of other changes,e.g. the inclusion of non nearest neighbour interactions and the consideration ofsymmetry under the left action of SO( m ) (frame indifference). As a result it isnatural to replace our assumption that the microscopic interaction is convex close toits minimum by a more complicated condition. We will thus address the applicationof our ideas to vector valued fields and models relevant in elasticity in future work. 1. INTRODUCTION Fourthly, in this work we focus on the behaviour of the partition function in thelarge volume limit. As in the work of Bauerschmidt, Brydges and Slade [ BBS15b ]it should be possible to study finer properties, e.g., correlation functions. As a firststep in that direction Hilger has recently shown that the scaling limit of the randomfield becomes a free Gaussian field on the torus (with the renormalised covariance)and that suitably averaged correlation functions converge in the infinite volumelimit [ Hil16 ].HAPTER 2 Setting and Results Let L > N we consider the space V N = { ϕ : Z d → R ; ϕ ( x + k ) = ϕ ( x ) ∀ k ∈ ( L N Z ) d } that can be identified with the set of functions on the torus T N = (cid:0) Z /L N Z (cid:1) d .Using | x | ∞ = max i =1 ,...,d | x i | for any x ∈ R d (reserving the notation | x | for theEuclidean norm pP x i ), the torus T N may be represented by the lattice cubeΛ N = { x ∈ Z d : | x | ∞ ≤ ( L N − } of side L N , once it is equipped with the metric ρ ( x, y ) = inf {| x − y + k | ∞ : k ∈ ( L N Z ) d } . We view V N as a Hilbert space with thescalar product ( ϕ, ψ ) = X x ∈ T N ϕ ( x ) ψ ( x ) . By X N we denote the subspace(2.1) X N = { ϕ ∈ V N : X x ∈ T N ϕ ( x ) = 0 } , of height fields whose sum over the torus is zero. We use λ N to denote the ( L Nd − X N . We equip the space X N with the σ -algebra B X N induced by the Borel σ -algebra with respect to the product topology anduse M ( X N ) = M ( X N , B X N ) to denote the set of probability measures on X N ,referring to elements in M ( X N ) as to random gradient fields .In this article we study a class of random gradient fields defined (as Gibbsmeasures) in terms of a non-convex perturbation of a Gaussian gradient field. Fora precise definition, we first introduce the discrete derivatives (2.2) ∇ i ϕ ( x ) = ϕ ( x + e i ) − ϕ ( x ) , ∇ ∗ i ϕ ( x ) = ϕ ( x − e i ) − ϕ ( x )on V N . Here, e i , i = 1 , . . . , d , are unit coordinate vectors in R d . Next, let E N ( ϕ )be the Dirichlet form(2.3) E N ( ϕ ) = 12 X x ∈ T N d X i =1 (cid:0) ∇ i ϕ ( x ) (cid:1) . Choosing a function V : R → R (satisfying the conditions to be specified later), weconsider the Gibbs mesure on the torus corresponding to the Hamiltonian(2.4) H N ( ϕ ) = E N ( ϕ ) + X x ∈ T N d X i =1 V ( ∇ i ϕ ( x )) . To be able to discuss random fields with a tilt u = ( u . . . , u d ) ∈ R d , we use themethod proposed by Funaki and Spohn [ FS97 ] who enforce the tilt on a measure defined on the torus space X N by replacing the gradient ∇ i ϕ ( x ) in all definitionsabove by ∇ i ϕ ( x ) − u i , i = 1 , . . . , d , x ∈ T N .Namely, we define the Gibbs mesure on T N at inverse temperature β as(2.5) γ uN,β (d ϕ ) = 1 Z N,β ( u ) exp (cid:0) − βH uN ( ϕ ) (cid:1) λ N (d ϕ ) , where(2.6) H uN ( ϕ ) = E N ( ϕ ) + 12 L Nd | u | + X x ∈ T N d X i =1 V ( ∇ i ϕ ( x ) − u i )(in the last equation we used the fact that substituting ∇ i ϕ ( x ) 7→ ∇ i ϕ ( x ) − u i in E N , the linear term P x ∈ T N P di =1 u i ∇ i ϕ ( x ) vanishes as P x ∈ T N ∇ i ϕ ( x ) = 0 foreach ϕ ∈ V N and each i = 1 , . . . , d ). Again, Z N,β ( u ) is the normalizing partitionfunction(2.7) Z N,β ( u ) = Z X N exp (cid:0) − βH uN ( ϕ ) (cid:1) λ N (d ϕ ) . Even though the ultimate goal, in general, is to characterize all limiting gradientGibbs measures with a fixed mean tilt, and, in particular cases, to prove theirunicity, in this paper we will restrict our attention to the discussion of the strictconvexity, in u , of the surface tension(2.8) σ β ( u ) := − lim N →∞ βL dN log Z N,β ( u ) . To state our main result, we need a condition on smallness of the perturbation V . We will state it in terms of the function K V,β,u : R d → R associated with theperturbation V : R → R determining the Hamiltonian H uN in (2.6) (and with the(inverse) temperature β ≥ u ∈ R d ). Namely, we take(2.9) K V,β,u ( z ) = exp (cid:8) − β d X i =1 U (cid:0) z i √ β , u i (cid:1)(cid:9) − U ( s, t ) = V ( s − t ) − V ( − t ) − V ′ ( − t ) s. First, we rewrite the partition function in terms of the function K V,β,u . Considerthe Gaussian measure ν β on X N corresponding to the Dirichlet form β E N ( ϕ ):(2.11) ν β (d ϕ ) = 1 Z (0) N,β exp (cid:0) − β E N ( ϕ ) (cid:1) λ N (d ϕ ) , with(2.12) Z (0) N,β = Z X N exp (cid:0) − β E N ( ϕ ) (cid:1) λ N (d ϕ ) . .2. MAIN RESULT 9 To avoid overloading of the notation, here and in future, we often skip the indexreferring to N (as above in the case of measure ν β ). Now, the partition function(2.7) is(2.13) Z N,β ( u ) = Z (0) N,β exp (cid:0) − β L Nd | u | (cid:1) Z X N exp (cid:0) − β X x ∈ T N d X i =1 V (cid:0) ∇ i ϕ ( x ) − u i (cid:1)(cid:1) ν β (d ϕ ) == Z (0) N exp (cid:0) − βL Nd ( | u | + V ( u )) (cid:1) Z X N exp (cid:0) − β X x ∈ T N d X i =1 U (cid:0) √ β ∇ i ϕ ( x ) , u i (cid:1)(cid:1) ν (d ϕ ) , where, denoting ν (d ϕ ) = ν β =1 (d ϕ ) and Z (0) N = Z (0) N,β =1 , the last equality was ob-tained by rescaling the field ϕ by √ β , invoking the definition (2.10) and using that P x ∈ T N ∇ i ϕ ( x ) = 0. Expanding the integrand(2.14) Y x ∈ T N (cid:16) (cid:8) − β d X i =1 U (cid:0) √ β ∇ i ϕ ( x ) , u i (cid:1)(cid:9) − (cid:17) above and introducing (with a slight abuse of notation), the function(2.15) K V,β,u ( X, ϕ ) = Y x ∈ X K V,β,u ( ∇ ϕ ( x ))for any subset X ⊂ T N , we get(2.16) Z N,β ( u ) = Z (0) N,β exp (cid:0) − βL Nd ( | u | + V ( u )) (cid:1) Z X N X X ⊂ T N K V,β,u ( X, ϕ ) ν (d ϕ ) . It will be useful to generalize our formulation slightly and, instead of a particu-lar K V,β,u above, to consider for each u a general function K u : R d → R and define(2.17) Z N ( u ) = Z X N X X K u ( X, ϕ ) ν (d ϕ )with(2.18) K u ( X, ϕ ) = Y x ∈ X K u ( ∇ ϕ ( x )) . Our main claim is that, under appropriate conditions on the function u 7→ K u , theperturbative component of the surface tension,(2.19) ς ( u ) := − lim N →∞ L dN log Z N ( u )is sufficiently smooth for small u .Before formulating it in detail, we observe that whenever the claim applies tothe case K u = K V,β,u , the uniform smoothness of ς ( u ) implies that, for sufficientlylarge β and small | u | , the surface tension σ ( u ) is strictly convex, since, in view of(2.16), we get(2.20) σ β ( u ) = | u | + V ( u ) + ς ( u ) β − lim N →∞ βL dN log Z (0) N,β The last term is a constant that does not depend on u . Given any ζ > 0, consider the Banach space E of functions K : R d → R withthe norm(2.21) kKk ζ = sup z ∈ R d X | α |≤ r ζ | α | (cid:12)(cid:12) ∂ α z K ( z ) (cid:12)(cid:12) e − ζ − | z | . Here, the sum is over nonnegative integer multiindices α = ( α , . . . , α d ), α i ∈ N , i = 1 , . . . , d with | α | = P di =1 α i ≤ r ∈ N , and ∂ α = Q di =1 ∂ α i i . We also use B δ (0) ⊂ R d to denote the ball B δ (0) = { u | | u | < δ } . Theorem Strict convexity of the surface tension ) . Let r ≥ . Thereexist constants δ > , ρ > , M > , and ζ > such that if the map R d ⊃ B δ (0) ∋ u 7→ K u ∈ E is C , satisfies the bounds (2.22) kK u k ζ ≤ ρ, and (2.23) d X i =1 (cid:13)(cid:13)(cid:13) ∂∂u i K u (cid:13)(cid:13)(cid:13) ζ + d X i,j =1 (cid:13)(cid:13)(cid:13) ∂ ∂u i ∂u j K u (cid:13)(cid:13)(cid:13) ζ + d X i,j,k =1 (cid:13)(cid:13)(cid:13) ∂ ∂u i ∂u j ∂u j K u (cid:13)(cid:13)(cid:13) ζ ≤ M with ζ ≥ ζ , ρ ≤ ρ , δ ≤ δ , M < M , and u ∈ B δ (0) , then the surface tension ς ( u ) exists with bounds on ς ( u ) , Dς ( u ) , D ς ( u ) , and D ς ( u ) depending only on ρ and M uniformly in u ∈ B δ (0) . The proof employs a multi-scale analysis based on ideas going back to the work[ BY90 ]. Even though we follow quite closely the approach outlined by Brydgesin [ Bry09 ], a fair amount of various deviations and generalisations is needed. Webelieve that this fact and the demands on clarity warrant an independent treatmentand the presentation of the proof in full detail.The reader familiar with [ Bry09 ] may, however, find various shortcuts. Tofacilitate a selective reading, we devote the next Chapter 3 to a presentation ofthe strategy of the proof, formulating then accurately all main steps of the proofand spelling out all needed extensions of [ Bry09 ] in Chapter 4. The proof is thenexecuted in full detail in the remaining chapters.Before passing to the outline of the proof, we discuss two particular classes ofperturbative potentials for which the above theorem applies.First we verify the assumptions of Theorem 2.1 for a class of perturbations ofthe form (2.9). This yields a very simple example of a possibly non-convex potentialat low temperatures. Proposition . Let r ∈ N , ζ ∈ (0 , ∞ ) , M ≥ , and suppose that (2.24) V ∈ C r +5 ( R ) , (2.25) V (0) = V ′ (0) = V ′′ (0) = 0 , (2.26) k D k V k ∞ ≤ M for ≤ k ≤ r + 5 , and (2.27) V ( s ) ≥ − ζ − s for each s ∈ R . .2. MAIN RESULT 11 Then, for any ρ ∈ (0 , / , there exists β = β ( ζ, ρ, M , r ) , δ = δ ( ζ, ρ, M , r ) ,and M ( ζ, M , r ) such that, for any β ≥ β , the map R d ⊃ B δ (0) ∋ u 7→ K V,β,u ∈ E is C and, for any u ∈ B δ (0) , (2.28) kK V,β,u k ζ ≤ ρ and (2.29) d X i =1 (cid:13)(cid:13)(cid:13) ∂ K V,β,u ∂u i (cid:13)(cid:13)(cid:13) ζ + d X i,j =1 (cid:13)(cid:13)(cid:13) ∂ K V,β,u ∂u i ∂u j (cid:13)(cid:13)(cid:13) ζ + d X i,j,k =1 (cid:13)(cid:13)(cid:13) ∂ K V,β,u ∂u i ∂u j ∂u j (cid:13)(cid:13)(cid:13) ζ ≤ M. Moreover, if r ≥ , there exists ¯ β ( M ) and ¯ δ ( M ) such that for all β ≥ ¯ β , thefunction σ β : B ¯ δ (0) → R given in (2.20) is C and uniformly strictly convex. The proof will be given in Section 2.3. Remark . (i) Notice that there is no loss of generality in the assumption(2.25). Indeed, the absolute term is just a shift by a constant, the linear termvanishes in view of the condition P x ∈ T N ∇ i ϕ ( x ) = 0, and the quadratic term maybe absorbed into the a priori quadratic part (2.3).(ii) The only smallness assumption on V is (2.27). In terms of the full macroscopicpotential W ( s ) = | s | + V ( s ) it reads(2.30) W ( s ) ≥ (cid:0) W ′′ (0) − ζ − (cid:1) s . Of course, the factor can be replaced by any θ < 1. If we could (almost) achievethe optimal value for ζ , ζ − = , the condition (2.30) would simply say that W isbounded from below by a nondegenerate quadratic function. Due to a number oftechnical points, however, we need to choose ζ − rather small to assure the validityof Theorem 2.1. ⋄ Another example is the non-convex potential considered in [ BK07 ]. The im-portance of this case lies in the fact that it is a non-convex potential for which thenon-uniqueness of a Gibbs state for a particular temperature and with a particu-lar tilt is actually proven. For the sake of simplicity, the potential considered in[ BK07 ] was chosen in a particular form that corresponds to the replacement ofexp (cid:8) − βH N ( ϕ ) (cid:9) by(2.31) Y x ∈ T N d Y i =1 h p exp n − (cid:0) ∇ i ϕ ( x ) (cid:1) o + (1 − p ) exp n − κ (cid:0) ∇ i ϕ ( x ) (cid:1) oi (for parameters κ O and κ D from [ BK07 ] we choose κ O = 1 and κ D = κ ). Thisamounts to replacing K V,β,u ( z ) = exp (cid:8) − β P di =1 V (cid:0) z i √ β − u i (cid:1)(cid:9) − K κ, p ,u ( z ) = d Y i =1 h p + (1 − p ) exp n 12 (1 − κ ) (cid:0) z i − u i ) oi − . Indeed, it is enough to observe that (2.31) can be rewritten as(2.33) exp (cid:8) −E N ( ϕ ) (cid:9) Y x ∈ T N d Y i =1 h p + (1 − p ) exp n − 12 (1 − κ ) (cid:0) ∇ i ϕ ( x ) (cid:1) oi . Notice that temperature β is in (2.31) and (2.32) is replaced by the parameter p .The phase transition (non-unicity of Gibbs state with the tilt u = 0) mentioned above happens, for κ sufficiently small, for a particular value p = p t ( κ ). However,this does not prevent the corresponding surface tension to be convex in u (at leastfor small | u | ) once p is sufficiently close to 1 (and thus bigger than p t ). Thiscorresponds to the condition of sufficiently large β in the previous Proposition.Observing that the map R d ∋ u 7→ K κ, p ,u ∈ E is clearly analytic for all p , whatonly needs to be proven to apply Theorem 2.1 is the following claim. Proposition . Let κ ∈ (0 , be given. There exist δ > , ζ = ζ ( δ ) and M so that so that for any | u | ≤ δ one has (2.34) kK κ, p ,u k ζ ≤ ρ and (2.35) d X i =1 (cid:13)(cid:13)(cid:13) ∂∂u i K κ, p ,u (cid:13)(cid:13)(cid:13) ζ + d X i,j =1 (cid:13)(cid:13)(cid:13) ∂ ∂u i ∂u j K κ, p ,u (cid:13)(cid:13)(cid:13) ζ + d X i,j,k =1 (cid:13)(cid:13)(cid:13) ∂ ∂u i ∂u j ∂u k K κ, p ,u (cid:13)(cid:13)(cid:13) ζ ≤ M for any − p sufficiently small (in dependence on ρ and ζ ). The proof is given below in Section 2.3 We collect the outstanding proofs for our two examples above. Proof of Proposition 2.2. Step 1. Estimate for kK V,β,u k ζ .This is the key estimate. The main idea is that for z i small (and also u i small)we can use the Taylor expansion of U ( z i √ β , u i ) in z i , while for large z i we rely onthe weight e − ζ − | z i | combined with the quadratic lower bound (2.27) on V .First, let us show that(2.36) − βU ( z i √ β , u i ) ≤ ζ − z i for any z i ∈ R and any | u | < δ, whenever δ ≤ M ζ − .Indeed, the Taylor expansion yields(2.37) β (cid:12)(cid:12)(cid:12) U ( z i √ β , u i ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12) V ′′ ( s ) (cid:12)(cid:12) z i with | s | ≤ | u i | + (cid:12)(cid:12) z i √ β (cid:12)(cid:12) . Since V ′′ (0) = 0 implies that (cid:12)(cid:12) V ′′ ( s ) (cid:12)(cid:12) ≤ M | s | , the righthand side is bounded by M (cid:0) δ + (cid:12)(cid:12) z i √ β (cid:12)(cid:12)(cid:1) z i yielding the claim for (cid:12)(cid:12) z i √ β (cid:12)(cid:12) ≤ δ .On the other hand, for (cid:12)(cid:12) z i √ β (cid:12)(cid:12) ≥ δ we use (2.27) and the observation that | a | ≥ | b | implies that ( a − b ) ≤ a to get(2.38) − βV ( z i √ β − u i ) ≤ ζ − z i . Moreover, expanding V ′ ( − u i ) around V ′ (0) = 0 up to the order u i , for | z i √ β | ≥ δ we get(2.39) β (cid:12)(cid:12)(cid:12) V ′ ( − u i ) z i √ β (cid:12)(cid:12)(cid:12) ≤ β M δ (cid:12)(cid:12)(cid:12) z i √ β (cid:12)(cid:12)(cid:12) ≤ M δz i .3. PROOFS OF THE GIVEN EXAMPLES 13 and, similarly,(2.40) β | V ( − u i ) | ≤ β M δ ≤ M δz i , yielding the claim since M ( + ) M < .As a result of (2.36), we are done once | z | = P di =1 z i ≥ ζ log ρ . Indeed,under this assumption, we have(2.41) (cid:12)(cid:12) e − βU ( zi √ β ,u i ) − (cid:12)(cid:12) e − ζ − | z | ≤ max (cid:0) e − βU ( zi √ β ,u i ) , (cid:1) e − ζ − | z | ≤ e − ζ − | z | ≤ ρ . Hence, we now focus on the case(2.42) | z | ≤ ζ log 2 ρ . For sufficiently small ρ , set(2.43) δ = ζ − M min (cid:0) , ρ ρ (cid:1) ≤ β = 2 ζ log ρ δ ≥ . Then, for β ≥ β , the relation (2.42) implies that | z | / √ β ≤ | z | / √ β ≤ δ and (2.37)thus for δ ≤ δ yields(2.45) β d X i =1 (cid:12)(cid:12)(cid:12) U ( z i √ β , u i ) (cid:12)(cid:12)(cid:12) ≤ M δ | z | ≤ ρ . Since | e t − | ≤ | t | for t ≤ 1, we get(2.46) (cid:12)(cid:12) e − β P di =1 U ( zi √ β ,u i ) − (cid:12)(cid:12) ≤ ρ . Together with (2.41) this shows that(2.47) sup z ∈ R d | e − β P di =1 U ( zi √ β ,u i ) − | e − ζ − | z | ≤ ρ | u | ≤ δ ≤ δ and β ≥ β with δ and β given by (2.43) and (2.44),respectively. Step 2. z -derivatives of K V,β,u .We will employ Fa`a di Bruno’s chain rule for higher order derivatives [ Har ] ofa function in the form e f ,(2.48) e − f ∂ α e f = X τ , τ ,...,m ,m ,... P j m j τ j = α α !( τ !) m ( τ !) m · · · m ! m ! · · · Y j ( ∂ τ j f ) m j . Here, the sum is over distinct partitions τ , τ , . . . of the multiindex α with mul-tiplicities m , m , . . . (i.e., such that P j m j τ j = α ) and τ ! = τ ! . . . τ d ! for anymultiindex τ = ( τ , . . . , τ d ).In our case, we have f ( z ) = − β P di =1 U ( z i √ β , u i ) with(2.49) ∂ z j f ( z ) = − p β (cid:0) V ′ ( z j √ β − u j ) − V ′ ( − u j ) (cid:1) . As for the higher derivatives, only the “diagonal” ones, ∂ kz j f ( z ), are non-vanishing,(2.50) ∂ kz j f ( z ) = − β ( k − / V ( k ) ( z j √ β − u j ) . For | u i | ≤ δ , we get(2.51) | ∂ z j f ( z ) | = (cid:12)(cid:12) V ′′ ( z j √ β − u j ) (cid:12)(cid:12) ≤ M min (cid:0) , δ + (cid:12)(cid:12) z j √ β (cid:12)(cid:12)(cid:1) and thus, using that ∂ z j f (0) = 0, also(2.52) | ∂ z j f ( z ) | ≤ M min (cid:0) , δ + (cid:12)(cid:12) z j √ β (cid:12)(cid:12)(cid:1) | z j | . Moreover, in view of (2.50), we have(2.53) sup | ∂ kz j f ( z ) | ≤ β ( k − / M for k ≥ 2. Combining (2.48) with (2.53) and with the particular implication of(2.52),(2.54) | ∂ z j f ( z ) | ≤ M | z | , observing that | z | r ≤ | z | r whenever r ≤ r , and using that M ≥ β ≥ (cid:12)(cid:12)(cid:12) ∂ α e − β P di =1 U ( zi √ β ,u i ) (cid:12)(cid:12)(cid:12) ≤ C ( r )e − β P di =1 U ( zi √ β ,u i ) M r (1 + | z | r )with a suitable constant C ( r ). Using, further, (2.36) and (2.55), we get (note that ζ ≥ ζ | α | (cid:12)(cid:12)(cid:12) ∂ α e − β P di =1 U ( zi √ β ,u i ) (cid:12)(cid:12)(cid:12) e − ζ − | z | ≤ ξ e − ζ − | z | with(2.57) ξ = ξ ( r , h, M ) = 2 C ( r ) ζ r M r (cid:0) r (cid:1) r / . Here, the factor ξ is a bound on the term C ( r ) ζ r M r e − ζ − | z | (1 + | z | r ) obtainedwith help of the identity max t> e − at t s = s s e − s a − s with t = | z | . As a result, theright hand side of (2.56) is bounded by ρ/ | z | ≥ ζ log ξρ .For(2.58) | z | ≤ ζ log 2 ξρ we take(2.59) δ = min (cid:0) δ , ζ − M log ξρ (cid:1) and(2.60) β = max (cid:8) β , ζ log ξρ δ (cid:9) . Then, for β ≥ β and | u | ≤ δ ≤ δ , the bound (2.58) implies that | z i |√ β ≤ δ , yielding,in view of (2.52), the estimate(2.61) | ∂ z j f ( z ) | ≤ M δ | z j | .3. PROOFS OF THE GIVEN EXAMPLES 15 and thus(2.62) | f ( z ) | ≤ d X j =1 M δ | z j | ≤ , again in view of (2.58) and the definition of δ . Hence, similarly as in (2.56), weget(2.63) (cid:12)(cid:12)(cid:12) ζ | α | ∂ α e − f ( z ) (cid:12)(cid:12)(cid:12) e − ζ − | z | ≤ C ( r ) ζ r e (2 M ) r | z | r e − ζ − | z | max (cid:0) δ , √ β (cid:1) ≤≤ C ( r , M , h ) max (cid:0) δ , √ β (cid:1) with C ( r , M , h ) = 2 C ( r ) ζ r e (2 M ) r (cid:0) r (cid:1) r . The factor max (cid:0) δ , √ β (cid:1) stemsfrom the fact that each first and second derivative of f contributes a factor boundedby 2 M δ (cf. (2.52) and (2.51)), while each higher derivative the factor boundedby M √ β (cf. (2.53)). Taking now(2.64) δ = min (cid:0) δ , ρC ( r ,M ,h ) (cid:1) and(2.65) β = max (cid:0) β , (cid:0) C ( r ,M ,h ) ρ (cid:1) (cid:1) , we get the sought claim(2.66) kK V,β,u k ζ ≤ ρ whenever | u | ≤ δ ≤ δ and β ≥ β . Step 3. u -derivatives of K V,β,u .The estimates for the u -derivatives of K V,β,u are similar. Indeed,(2.67) ∂ u i K V,β,u = e f ( z ) ∂ i f ( z ) , (2.68) ∂ u j ∂ u i K V,β,u = e f ( z ) (cid:0) f j ( z ) f i ( z ) − f i,j ( z ) (cid:1) , etc., where(2.69) f i ( z ) = − β d X i =1 U (1) ( z i √ β , u i ) , (2.70) f i,i ( z ) = − β d X i =1 U (2) ( z i √ β , u i ) , and f i,j ( z ) = 0 if i = j. Here, the functions U ( ℓ ) have the same structure as U , but with V replaced by( − ℓ ∂ ℓ V , e.g.,(2.71) U (1) ( s, t ) = V ′ ( s − t ) − V ′ ( − t ) − V ′′ ( − t ) s. Thus, as in (2.53) and (2.54), we get(2.72) β sup | ∂ kz j U ( ℓ ) ( z i √ β , u i ) | ≤ sup | ∂ k + ℓ V | ≤ M and(2.73) β | ∂ z j U ( ℓ ) ( z i √ β , u i ) | ≤ sup | ∂ ℓ V || z i | ≤ M | z i | . In addition, we have a new estimate(2.74) β | U ( ℓ ) ( z i √ β , u i ) | ≤ sup | ∂ ℓ V || z i | ≤ M | z i | . Thus, for | β | ∈ { , , } , | α | ∈ { , . . . , r } ,(2.75) β (cid:12)(cid:12) ∂ α z ∂ β u e − β P di =1 U ( zi √ β ,u i ) (cid:12)(cid:12) ≤ C ( r )e f ( z ) M | α | + | β | (1 + | z | ) | α | + | β | . Estimate (2.36) yields | f ( z ) | ≤ ζ − | z | if | u | ≤ M ζ − (in particular if | u | < δ defined in (2.64)). Then we easily conclude that(2.76) (cid:13)(cid:13)(cid:13) ∂ β K V,β,u (cid:13)(cid:13)(cid:13) ζ ≤ M ( r , h, M ) for any | β | ∈ { , , } , | u | ≤ δ , and β ≥ β , with a suitable M ( r , h, M ). Step 4. Uniform convexity of σ ( u ).To obtain uniform convexity of σ ( u ), we first fix ρ so small and r and ζ solarge that Theorem 2.1 applies. Then for β ≥ β and | u | < δ we find that ς ( u ) isa C function and its first three derivatives in B δ (0) are controlled in terms of ρ and M = M ( r , h, M ). In particular,(2.77) | D ς ( u ) | ≤ M ′ ( ζ, M , ρ ) if u ∈ B δ (0) . Note that for | s | ≤ M , we have V ′′ ( s ) ≥ − . Let(2.78) ¯ δ ( M ) = min (cid:0) δ ( ζ, M , ρ, r ) , M (cid:1) and(2.79) ¯ β ( M ) = max (cid:0) β ( ζ, M , ρ, r ) , M ′ ( ζ, M , r ) (cid:1) . Then(2.80) D σ ( u ) ≥ Id − Id − Id ≥ Idfor u ∈ B ¯ δ (0) and β ≥ ¯ β . (cid:3) Proof of Proposition 2.4. The proof is similar as the proof of Proposition 2.2.We will only indicate the main steps. Again, skipping the indices in K κ, p ,u andrewriting(2.81) K ( z ) = d Y i =1 " − p ) h exp n 12 (1 − κ ) (cid:0) z i − u i ) o − i − , we have(2.82) 0 ≤ K ( z ) ≤ d (1 − p ) exp n d X i =1 (cid:0) z i − u i ) o , and, with suitable polynomials P α ( z − u ), also(2.83) |∇ α K ( z ) | ≤ (1 − p ) P α ( z − u ) exp n d X i =1 (cid:0) z i − u i ) o . Taking now sufficiently small u and, then, sufficiently large ζ we have .3. PROOFS OF THE GIVEN EXAMPLES 17 kKk ζ ≤ C (1 − p )with the constant C depending on ζ . Similar bounds are valid for the remainingterms in (2.35). (cid:3) HAPTER 3 The Strategy of the Proof Here we present, in rather broad brush, the main ideas of the proof. Accuratedefinitions of the needed notions then follow in the succeeding chapter.As mentioned above, to verify the claim of the theorem, we need to prove thatthe finite volume perturbative component of the surface tension(3.1) ς N ( u ) := − L dN log Z N ( u )has bounded derivatives uniformly in N ∈ N .Here, the partition function Z N ( u ) can be expressed, with a flavour of clusterexpansions, in terms of the functions K ( X, ϕ ) = K u ( X, ϕ ) as shown in (2.17).However, here comes a difficulty: even though the function K ( X, ϕ ) depends onlyon ϕ ( x ) with x in the set X and its close neighbourhood and even if for a disjointunion X = X ∪ X one has K ( X, ϕ ) = K ( X , ϕ ) K ( X , ϕ ), the Gaussian measure ν (d ϕ ) with its slowly decaying correlations does not allow to separate the integralof K ( X, ϕ ) into a product of integrals with the integrands K ( X , ϕ ) and K ( X , ϕ ).This is a non-locality that has to be overcome.The strategy is to perform the integration in steps corresponding to increasingscales. Before showing what we mean by that, let us make one simple modification.Its importance will be in providing a parameter that will allow us to fine-tune theprocedure in such a way that the final integration will eventually yield a result witha straightforward bound.The parameter in question will be chosen as a symmetric d × d -matrix q ∈ R d × d sym .Multiplying and dividing the integrand in (2.17) by(3.2) exp n − X x ∈ T N d X i,j =1 q i,j ∇ i ϕ ( x ) ∇ j ϕ ( x ) o = exp n − X x ∈ T N h q ∇ ϕ ( x ) , ∇ ϕ ( x ) i o and using the definition of the measure ν (by (2.11) with β = 1), we get(3.3) Z N ( u ) = Z ( q ) N Z (0) N Z X N exp n − X x ∈ T N h q ∇ ϕ ( x ) , ∇ ϕ ( x ) i o X X K ( X, ϕ ) µ ( q ) (d ϕ ) . Here, µ ( q ) is the Gaussian measure on X N with the Green function C ( q ) , the inverseof the operator A ( q ) = P di,j =1 (cid:0) δ i,j − q i,j (cid:1) ∇ ∗ i ∇ j ,(3.4) µ ( q ) (d ϕ ) = exp (cid:8) −E q ( ϕ ) (cid:9) λ N (d ϕ ) Z ( q ) N , with(3.5) E q ( ϕ ) = ( A ( q ) ϕ, ϕ ) = X x ∈ T N d X i,j =1 (cid:0) δ i,j − q i,j (cid:1) ∇ i ϕ ( x ) ∇ j ϕ ( x ) , 190 3. THE STRATEGY OF THE PROOF and(3.6) Z ( q ) N = Z X N exp (cid:8) −E q ( ϕ ) (cid:9) λ N (d ϕ ) . Under a suitable assumption about the smallness of q (so that, in particu-lar, the matrix − q is positive definite), we will show that the Gaussian mea-sure µ ( q ) can be decomposed into a convolution µ ( q ) (d ϕ ) = µ ( q )1 ∗ · · · ∗ µ ( q ) N +1 (d ϕ )where µ ( q )1 , . . . , µ ( q ) N +1 are Gaussian measures with a particular finite range property.Namely, the covariances C ( q ) k ( x ) of the measures µ ( q ) k , k = 1 , . . . , N + 1, vanish for | x | ≥ L k with a fixed parameter L with an additional bound on their derivativeswith respect to q of the order L − ( k − d − . (See next Chapter for careful definitionsand exact formulations; here we concentrate just on the main ideas.)Now, let us write the integral in (3.3) symbolically as(3.7) Z X N (e − H ( q ) ◦ K ( q ) )( ϕ ) µ ( q ) (d ϕ ) . Here(3.8) H ( q ) ( X, ϕ ) = X x ∈ X d X i,j =1 q i,j ∇ i ϕ ( x ) ∇ j ϕ ( x ) = X x ∈ X h q ∇ ϕ ( x ) , ∇ ϕ ( x ) i , the function K ( q ) is defined as(3.9) K ( q ) ( X, ϕ ) = exp n − X x ∈ X h q ∇ ϕ ( x ) , ∇ ϕ ( x ) i o K ( X, ϕ ) , and ◦ is the circle product notation for the convolutive sum over subsets X ⊂ T N ,(3.10) (e − H ( q ) ◦ K ( q ) )( ϕ ) = X X ⊂ T N e − H ( q ) ( T N \ X,ϕ ) K ( q ) ( X, ϕ ) , where we set H ( q ) ( ∅ , ϕ ) = K ( q ) ( ∅ , ϕ ) = 1.Replacing µ ( q ) in (3.7) by the convolution µ ( q )1 ∗ · · · ∗ µ ( q ) N +1 (d ϕ ), we will proceedby integrating first over µ ( q )1 . It turns out that the form of the integral is conserved.Namely, starting from H ( q )0 = H ( q ) and K ( q )0 = K ( q ) , we can define H ( q )1 and K ( q )1 so that(3.11) Z X N (e − H ( q )0 ◦ K ( q )0 )( ϕ + ξ ) µ ( q )1 (d ξ ) = (e − H ( q )1 ◦ K ( q )1 )( ϕ ) . Here, the function K ( q )1 ( X, ϕ ) is defined (nonvanishing) only for sets X consisting of L d -blocks and H ( q )1 is again a quadratic form like H ( q )0 but with modified coefficients q i,j and additional linear and constant terms. Recursively, one can define a sequenceof pairs ( H ( q )1 , K ( q )1 ) , ( H ( q )2 , K ( q )2 ) , . . . , ( H ( q ) N , K ( q ) N ) with each H ( q ) k a quadratic formin ∇ ϕ (plus linear and constant terms) and K ( q ) k ( X, ϕ ) defined for sets X consistingof L kd -blocks so that(3.12) Z X N (e − H ( q ) k ◦ K ( q ) k )( ϕ + ξ ) µ ( q ) k +1 (d ξ ) = (e − H ( q ) k +1 ◦ K ( q ) k +1 )( ϕ ) . Of course, the difficulty lies in producing correct definitions of consecutive pairsof functions H ( q ) k , K ( q ) k so that not only (3.12) is valid, but also that the form ofthe quadratic function H k is conserved, the coarse-grained dependence of K ( q ) k on . THE STRATEGY OF THE PROOF 21 blocks L dk is maintained, and, most importantly, the size of the perturbation K ( q ) k in a conveniently chosen norm decreases (the variable K ( q ) k is irrelevant in the lan-guage of the renormalisation group theory). See Propositions 4.3-4.6 for an explicitform and properties of the renormalisation transformation T ( q ) k : ( H ( q ) k , K ( q ) k ) ( H ( q ) k +1 , K ( q ) k +1 ).Using now sequentially the formula (3.12), we eventually get(3.13) Z X N (e − H ( q )0 ◦ K ( q )0 )( ϕ ) µ ( q ) (d ϕ ) = Z X N (e − H ( q ) N ◦ K ( q ) N )( ϕ ) µ ( q ) N +1 (d ϕ )and thus(3.14) Z N ( u ) = Z ( q ) N Z (0) N Z X N (e − H ( q ) N ◦ K ( q ) N )( ϕ ) µ ( q ) N +1 (d ϕ ) . At this moment we will invoke an additional feature. Namely, the finite rangedecomposition can be constructed in such a way that the measures µ ( q ) , . . . , µ ( q ) N +1 depend smoothly on q ([ AKM13 ]). As a result it turns out that, in dependence onthe original perturbation K u (or on V , β , and u in the explicit choice of K u as in(2.9)), one can choose the initial value q = q ( K u ) by an implicit function theoremin such a way that H ( q ) N = 0.However, here we encounter a difficulty stemming from the fact that the actionof T ( q ) k , considered on a scale of function spaces, depends on q with certain loss ofregularity, see Chapter 6. This leads to a need for employing a suitable version ofimplicit function theorem as well as a theorem about chain rule for composed mapswith loss of regularity (see Appendices D and E for the definitions and proofs).Also, the “starting” Hamiltonian H ( q )0 will in general contain, in addition tothe quadratic term given by (3.8), also linear and constant terms, i.e., H ( q )0 ( X, ϕ ) = P x ∈ X H ( x, ϕ ) with(3.15) H ( x, ϕ ) = λ + d X i =1 a i ∇ ϕ ( x ) + d X i,j =1 c i,j ∇ i ∇ j ϕ ( x ) + 12 d X i,j =1 q i,j ∇ ϕ ( x ) ∇ j ϕ ( x ) , see (4.91) and (4.17). Note, however, that the constant and linear terms do not leadto a change of the measure µ ( q ) since by periodicity of ϕ we have P x ∈ T N ∇ i ϕ ( x ) = 0and P x ∈ T N ∇ i ∇ j ϕ ( x ) = 0. For the purpose of this broad outline of the proof wewill pretend that we can achieve H ( q ) N = 0 with the choice λ = a = c = 0 . The general situation will be discussed in Chapter 4.5 below.Finally, taking into account that the function K ( q ) N ( X, · ) is defined only for X = Λ N or X = ∅ , we get(3.16) Z N ( u ) = Z ( q ) N Z (0) N Z X N (cid:0) K ( q ) N (Λ N , ϕ ) (cid:1) µ ( q ) N +1 (d ϕ ) , with q being implicitly dependent on K = K u by the condition that the iterationdescribed above gives H ( q ) N = 0. Note that this formula was derived under theassumption that the constant term λ in the initial perturbation is zero. In general,there is an additional term depending on λ , see (4.95) or (4.110). Now, to get the sought smoothness with respect to u , we have to evaluate thederivatives with respect to q and show the smooth dependence of implicitly defined q as function of u . The smoothness with respect to q is quite straightforward as thefactor Z ( q ) N can be explicitly computed by Gaussian integration and the derivativesof the integral term can easily be bounded as a consequence of the iterative boundson K ( q ) N . The smoothness of q as function of u follows by a careful examination ofthe corresponding implicit function yielding q as function of the initial perturbation K u and by smoothness of K u as function of u assumed in Theorem 2.1 and provenfor the particular classes of potentials considered in Propositions 2.2 and 2.4, seeChapter 4.6.HAPTER 4 Detailed Setting of the Main Steps First, we formulate the needed claim about the finite range decomposition ofthe Green function C ( q ) , the inverse of the operator A ( q ) = P di,j =1 (cid:0) δ i,j − q i,j (cid:1) ∇ ∗ i ∇ j on X N . We use k q k to denote the operator norm of q viewed as operator on R d equipped with ℓ metric. Obviously, k q k ≤ (cid:0)P i,j q i,j (cid:1) / . Proposition . Let q ∈ R d × d sym be a symmetric d × d -matrix such that k q k ≤ .There exist positive definite operators C ( q ) k , k = 1 , . . . , N + 1 , on X N such that (4.1) C ( q ) = N +1 X k =1 C ( q ) k . The operators C ( q ) k commute with translations on T N . In particular, there existsa function C ( q ) k on T N such that (cid:0) C ( q ) k ϕ (cid:1) ( x ) = P y ∈ T N C ( q ) k ( x − y ) ϕ ( y ) for each ϕ ∈ X N . Moreover, (4.2) C ( q ) k ( x ) = 0 if | x | ∞ ≥ L k and, for each multiindex α with | α | ≤ and any a ∈ N there exists a constant c α ,a such that (4.3) sup k q k≤ |∇ α D a C ( q ) k ( x )( ˙ q , . . . , ˙ q ) | ≤ c α ,a L − ( k − d − | α | ) L η ( | α | ,d ) k ˙ q k a for all x ∈ T N and all k = 1 , . . . , N + 1 , with (4.4) η ( n, d ) = max( ( d + n − , d + n + 6) + 10 . Here, ∇ α = Q di =1 ∇ α i i and D is the directional derivative in the direction ˙ q . The proof can be found in [ AKM13 ] which is an extension of ideas in [ BT06 ]and [ BGM04 ] applied to families of gradient Gaussian measures including vectorvalued functions. In fact there it is shown that C ( q ) k is (real) analytic in q with thenatural estimates for all derivatives with respect to q . Remark . Since the C ( q ) k are translation invariant they are diagonal in theFourier basis given by f p ( x ) = L − dN/ e i h p,x i with(4.5) p ∈ b T N = n p = ( p , . . . , p d ) : p i ∈ (cid:8) − ( L N − πL N , − ( L N − πL N . . . , , . . . , ( L N − πL N (cid:9)o , i.e.,(4.6) C ( q ) k f p = b C ( q ) k ( p ) f p , 234 4. DETAILED SETTING OF THE MAIN STEPS where the Fourier multiplier b C ( q ) k ( p ) is just the discrete Fourier transform of thekernel C ( q ) k . Equation (4.62) and Lemma 4.3 in [ AKM13 ] yield(4.7) 1 L dN X p ∈ b T N \{ } | p | n | D aq b C ( q ) k ( p )( ˙ q , . . . , ˙ q ) | ≤ a a ! c ( n, d ) L η ( n,d ) L − ( k − d + n − . This estimate implies (4.3) by the discrete Fourier inversion formula, but it willalso be of independent use later. ⋄ Now, if a random field ϕ is distributed with respect to the Gaussian measure µ ( q ) = µ C ( q ) on X N , where the covariance C ( q ) admits a finite range decomposition(4.1), then there exist N + 1 independent random fields ξ k , k = 1 , . . . , N + 1, suchthat each ξ k is distributed according to the Gaussian measure µ ( q ) k = µ C ( q ) k withthe covariance C ( q ) k and, in distribution,(4.8) ϕ = N +1 X k =1 ξ k , or,(4.9) Z X N F ( ϕ ) µ ( q ) (d ϕ ) = E N +1 · · · E F, where E k , k = 1 , . . . , N + 1, denote the expectations with respect to the Gaussianmeasures µ ( q ) k and F is taken as a function of P N +1 k =1 ξ k .Taking into account that operators C ( q ) k are of full rank on X N , standard Gauss-ian calculus yields an expression in terms of convolutions,(4.10) Z X N F ( ϕ ) µ ( q ) (d ϕ ) = Z X N F ( ϕ ) µ ( q )1 ∗ · · · ∗ µ ( q ) N +1 (d ϕ ) == Z X N ×···× X N F (cid:16) N +1 X k =1 ξ k (cid:17) µ ( q )1 (d ξ ) . . . µ ( q ) N +1 (d ξ N +1 ) . Our preferred formulation is to introduce renormalisation maps R ( q ) k on func-tions on X N by(4.11) ( R ( q ) k F )( ϕ ) = Z X N F ( ϕ + ξ ) µ ( q ) k (d ξ ) , k = 1 , . . . , N. Just to be on a firm ground, we can introduce the spaces M ( X N ) of all func-tions measurable with respect to λ N on X N and view R ( q ) k as a map R ( q ) k : U ⊂ M ( X N ) → M ( X N ), where U = { F : X N → R : r.h.s of (4.11) exists and is finite } . The integration R X N F ( ϕ ) µ ( q ) (d ϕ ) can be viewed, for any F ∈ M ( X N ), as theconsecutive application of maps R ( q ) k with a final integration with respect to µ ( q ) N +1 :(4.12) Z X N F ( ϕ ) µ ( q ) (d ϕ ) = Z X N ( R ( q ) N . . . R ( q )1 F )( ϕ ) µ ( q ) N +1 (d ϕ ) . Notice that for the operators C ( q ) N and C ( q ) N +1 (and the measures µ ( q ) N and µ ( q ) N +1 ) thecondition (4.2) is void. However, the suppression condition (4.3) still applies. .2. POLYMERS, POLYMER FUNCTIONALS, IDEAL HAMILTONIANS AND NORMS. 25 There is a natural hierarchical paving corresponding to the correlation range(4.2) of random fields governed by Gaussian measures µ k .Namely, for k = 0 , , , . . . , N , we pave the torus Λ N by L ( N − k ) d disjoint cubesof side length L k . These cubes are all translates ( L is odd) of { x ∈ Λ N : | x | ∞ ≤ ( L k − } by vectors in L k Z d . We call such cubes k -blocks or blocks of k -thgeneration, and use B k to denote the set of all k -blocks, B k = B k (Λ N ) = { B : B is a k -block } , k = 0 , , . . . , N. Single vertices of the lattice are 0-blocks, the starting generation for the renor-malisation group transforms, B = Λ N . The only N -block is the torus Λ N itself, B N = { Λ N } .A union of k -blocks is called a k -polymer . We use P k = P k (Λ N ) to denote theset of all k -polymers in Λ N and we have ∅ ∈ P k . As N is fixed through the majorpart of the paper, we often skip Λ N from the notation as indicated above. Noticethat certain ambiguity stems from the fact that every k -polymer is also j -polymerfor any j ≤ k . Nevertheless, we abstain from introducing k -polymer as a pair ( X, k )consisting of a set X (union of k -blocks) and a label; the appropriate label will bealways clear from the context.Any subset X ⊂ T N is said to be connected if for any x, y ∈ X there exista path x = x, x , . . . , x n = y such that | x i +1 − x i | ∞ = 1, i = 1 , . . . , n − 1. Weuse C ( X ) to denote the set of connected components of X . Two connected sets X, Y ⊂ Λ N are said to be strictly disjoint if their union is not connected. Noticethat for any strictly disjoint X, Y ∈ P k , we have dist( X, Y ) > L k .We use P c k to denote the set of all connected k -polymers and we define that ∅ / ∈ P c k . For a polymer X ∈ P k , we use B k ( X ) to denote the set of k -blocksin X and | X | k = |B k ( X ) | to denote the number of k -blocks in X and P k ( X ) todenote the set of all polymers Y consisting of subsets of blocks from B k ( X ). Theset difference X \ Y ∈ P k of two polymers X, Y ∈ P k is again a polymer from P k , X \ Y = ∪ B ∈ X,B / ∈ Y B . The closure X of a polymer X ∈ P k is the smallest polymer Y ∈ P k +1 of the next generation such that X ⊂ Y .A polymer X ∈ P c k is called small if | X | k ≤ d and we denote S k = { X ∈P c k : | X | k ≤ d } . For any B ∈ B k we define its small set neighbourhood B ∗ to bethe cube of the side (2 d +1 − L k centered at B . Notice that B ∗ is the smallestcube for which B ⊂ Y and Y ∈ S k implies Y ⊂ B ∗ . For any polymer X ∈ P k weuse X ∗ to denote its small set neighbourhood , X ∗ = ∪{ B ∗ : B ∈ B k ( X ) } . Noticethat, strictly speaking, the operation of closure X and small set neighbourhood X ∗ should be amended by an index k + 1 or k indicating the scale from which therelevant blocks are taken. Again we will abstain from cumbersome indexing andavoid ambiguity by clearly stating to which P k the considered set X is taken tobelong.Having fixed the parameter N and using a shorthand X for X N in the following,we first introduce the space M ( P k , X ) of all maps F : P k × X → R such that forall X ∈ P k one has F ( X, · ) ∈ M ( X ), the map F is L k -periodic ( F ( τ a ( X ) , τ a ( ϕ )) = F ( X, ϕ ) for any a ∈ ( L k Z ) d , where τ a ( B ) = B + a and τ a ( ϕ )( x ) = ϕ ( x − a ))and F ( X, ϕ ) depends only on values of ϕ on X ∗ ( ϕ, ψ ∈ X , ϕ (cid:12)(cid:12) X ∗ = ψ (cid:12)(cid:12) X ∗ = ⇒ F ( X, ϕ ) = F ( X, ψ ) with ϕ (cid:12)(cid:12) X ∗ denoting the restriction of ϕ to X ∗ ). The sets M ( P c k , X ) , M ( S k , X ), and M ( B k , X ) are defined in an analogous way.We also consider the set M ∗ ( B k , X ) ⊃ M ( B k , X ) of the maps F : B k × X → R with F ( B, ϕ ) depending only on values of ϕ on the extended set ( B ∗ ) ∗ .For functions from M ( P k , X ) we introduce the circle product ,(4.13) F , F ∈ M ( P k , X ) , ( F ◦ F )( X, ϕ ) = X Y ⊂ X F ( Y, ϕ ) F ( X \ Y, ϕ ) , where we defined F ( ∅ , ϕ ) =: 1. Notice, that the product is defined pointwise inthe variable ϕ . We often skip it and write ( F ◦ F )( X ) = P Y ⊂ X F ( Y ) F ( X \ Y ).Observe that the circle product is commutative and distributive.For F ∈ M ( B k , X ) and X ∈ P k , we define(4.14) F X ( ϕ ) = Y B ∈B k ( X ) F ( B, ϕ ) . Extending any F ∈ M ( B k , X ) to M ( P k , X ) by taking(4.15) F ( X, ϕ ) = F X ( ϕ ) , we get(4.16) ( F + F ) X = X Y ⊂ X F Y F X \ Y = ( F ◦ F )( X )directly from the definitions.For each x ∈ Λ N we define the functions(4.17) H ( x, ϕ ) = λ + d X i =1 a i ∇ ϕ ( x ) + d X i,j =1 c i,j ∇ i ∇ j ϕ ( x ) + 12 d X i,j =1 d i,j ∇ ϕ ( x ) ∇ j ϕ ( x )with coefficients λ ∈ R , a ∈ R d , c ∈ R d × d and d ∈ R d × d sym .A special role will be played by a subspace M ( B k , X ) ⊂ M ( B k , X ) of allquadratic functions built from (4.17) of the form(4.18) H ( B, ϕ ) = X x ∈ B H ( x, ϕ ) = λ | B | + ℓ ( ϕ ) + Q ( ϕ ) , where(4.19) ℓ ( ϕ ) = X x ∈ B (cid:2) d X i =1 a i ∇ i ϕ ( x ) + d X i,j =1 c i,j ∇ i ∇ j ϕ ( x ) (cid:3) and(4.20) Q ( ϕ, ϕ ) = 12 X x ∈ B d X i,j =1 d i,j ∇ i ϕ ( x ) ∇ j ϕ ( x ) . Sometimes we use the term ideal Hamiltonians for functions in M ( B k , X ).Our next aim is to introduce norms k·k k,r and k·k k +1 ,r on M ( P k , X ) (with r = 1 , . . . , r , where r is a fixed integer to be chosen later) and a norm k·k k, on M ( B k , X ). We begin by introducing, for each k ∈ { , , . . . , N } and X ∈ P k , twodistinct (semi)norms | · | k,X and | · | k +1 ,X on X . For any ϕ ∈ X we define(4.21) | ϕ | k,X = max ≤ s ≤ sup x ∈ X ∗ h L k (cid:0) d − 22 + s (cid:1)(cid:12)(cid:12) ∇ s ϕ ( x ) (cid:12)(cid:12) .2. POLYMERS, POLYMER FUNCTIONALS, IDEAL HAMILTONIANS AND NORMS. 27 and(4.22) | ϕ | k +1 ,X = max ≤ s ≤ sup x ∈ X ∗ h L ( k +1) (cid:0) d − 22 + s (cid:1)(cid:12)(cid:12) ∇ s ϕ ( x ) (cid:12)(cid:12) , where(4.23) |∇ s ϕ ( x ) | = X | α | = s |∇ α ϕ ( x ) | . Next, for any s -linear function S k on X × · · · × X , we define(4.24) | S | j,X = sup | ˙ ϕ | j,X ≤ (cid:12)(cid:12) S k ( ˙ ϕ, . . . , ˙ ϕ ) (cid:12)(cid:12) , j = k, k + 1 , and, for any F ∈ C r ( X ), also(4.25) | F ( ϕ ) | j,X,r = r X s =0 s ! | D s F ( ϕ ) | j,X . Here, for s = 0 we take(4.26) | D F ( ϕ ) | j,X = | F ( ϕ ) | . In particular, considering for any F ∈ M ( P k , X ) and any X ∈ P k (and similarlyalso for any F ∈ M ( B k , X )) the map F ( X ) : X → R defined by F ( X )( ϕ ) = F ( X, ϕ )and its s th derivative D s F ( X, ϕ )( ˙ ϕ, . . . , ˙ ϕ ), we get(4.27) | F ( X, ϕ ) | j,X,r = r X s =0 s ! sup | ˙ ϕ | j,X ≤ (cid:12)(cid:12) D s F ( X, ϕ )( ˙ ϕ, . . . , ˙ ϕ ) (cid:12)(cid:12) , j = k, k + 1 . Now, we are ready to introduce the weighted strong norm |k F ( X ) k| k,X as wellas weighted weak norm k F ( X ) k k,X,r , r = 1 , . . . , r depending on parameters h and ω that will be used for tuning their properties. Introducing the strong weightfunctions(4.28) W Xk ( ϕ ) = exp n X x ∈ X G k,x ( ϕ ) o with(4.29) G k,x ( ϕ ) = 1 h (cid:0) |∇ ϕ ( x ) | + L k |∇ ϕ ( x ) | + L k |∇ ϕ ( x ) | (cid:1) , we define the weighted strong norm(4.30) |k F ( X ) k| k,X = sup ϕ | F ( X, ϕ ) | k,X,r W − Xk ( ϕ )with W − Xk ( ϕ ) = (cid:0) W Xk ( ϕ ) (cid:1) − . For F ∈ M ( B k , X ), the norm |k F ( B ) k| k,B actuallydoes not depend on B in view of periodicity of F , and we use the shorthand |k F k| k .Further, let B x ∈ B k be the k -block containing x and let ∂X denote the bounda-ry(4.31) ∂X = { y X | ∃ z ∈ X such that | y − z | = 1 }∪{ y ∈ X | ∃ z X such that | y − z | = 1 } (recall that |·| is the Euclidean norm). Introducing the weak weight functions(4.32) w Xk ( ϕ ) = exp n X x ∈ X ω (cid:0) d g k,x ( ϕ ) + G k,x ( ϕ ) (cid:1) + L k X x ∈ ∂X G k,x ( ϕ ) o with G k,x ( ϕ ) as above and(4.33) g k,x ( ϕ ) = 1 h X s =2 L (2 s − k sup y ∈ B ∗ x |∇ s ϕ ( y ) | , we define the weighted weak norm by(4.34) k F ( X ) k k,X,r = sup ϕ | F ( X, ϕ ) | k,X,r w − Xk ( ϕ ) , r = 1 , . . . , r . In addition we also introduce the norm k·k k : k +1 ,X,r that can be viewed as being“halfway between” k·k k,X,r and k·k k +1 ,U,r with U = X ∈ P k +1 . Namely, we define(4.35) k F ( X ) k k : k +1 ,X,r = sup ϕ | F ( X, ϕ ) | k +1 ,X,r w − Xk : k +1 ( ϕ ) , r = 1 , . . . , r . with(4.36) w Xk : k +1 ( ϕ ) = exp n X x ∈ X (cid:0) (2 d ω − g k : k +1 ,x ( ϕ ) + ωG k,x ( ϕ ) (cid:1) + 3 L k X x ∈ ∂X G k,x ( ϕ ) o , where(4.37) g k : k +1 ,x ( ϕ ) = 1 h X s =2 L (2 s − k +1) sup y ∈ B ∗ x |∇ s ϕ ( y ) | , Notice that for the functions g k : k +1 ,x entering the norm k·k k : k +1 ,X,r , we still takesup y ∈ B ∗ x with k -block B x . The prefactors L (2 s − k +1) , however, involve the power k + 1. Also, the norm | F ( X, ϕ ) | k +1 ,X,r is used, involving ˙ ϕ k +1 ,X in its definition.For any r ≤ r , clearly,(4.38) k F ( X ) k k,X,r ≤ |k F ( X ) k| k,X . Inspecting the definitions, it is also easy to show that(4.39) k F ( X ) k k : k +1 ,X,r ≤ k F ( X ) k k,X,r once ω ≥ d − (assuring that 2 d ω ( L − ≥ L ), and, for any U ∈ P k +1 ⊂ P k and F ∈ M ( P k +1 , X ) ⊂ M ( P k , X ), also(4.40) k F ( U ) k k +1 ,U,r ≤ k F ( U ) k k : k +1 ,U,r ≤ k F ( U ) k k,U,r . Next, for any F ∈ M ( P c k , X ) and a parameter A ∈ R + we introduce(4.41) k F k ( A ) k,r = sup X ∈P c k k F ( X ) k k,X,r Γ k, A ( X ) , r = 1 , . . . , r , where(4.42) Γ k, A ( X ) = ( A | X | if X ∈ P c k \ S k X ∈ S k . Similarly we define also k F k ( A ) k : k +1 ,r . Note that this norm is only defined via func-tional on connected polymers. Whenever we estimate functionals on arbitrary poly-mers we simply consider the product over the connected components. Occasionally,when the parameter A is clear from the context, we skip it and write just k F k k,r and k F k k : k +1 ,r . For F ∈ M ( B k , X ) we also define(4.43) k F k (b) k,r = k F ( B ) k k,B,r . .3. RENORMALISATION TRANSFORMATION T k : ( H k , K k ) ( H k +1 , K k +1 ) 29 Notice that the right hand side does not depend on B in view of L k -periodicity of F . Any F ∈ M ( P k , X ) can be restricted to M ( B k , X ) with k F k (b) k,r ≤ k F k k,r .Finally, on the subspace M ( B k , X ) we define an additional norm k·k k, bytaking(4.44) k H k k, = L dk | λ | + L dk h d X i =1 | a i | + L ( d − k h d X i,j =1 | c i,j | + h d X i,j =1 | d i,j | for any H ∈ M ( B k , X ) of the form (4.18).Also, let us stress that the above norms depend on parameters like L , h , and A that are often skipped from the notation. Finally we use the notation(4.45) M k,r := { K ∈ M ( P c k , X ) : k K k ( A ) k,r < ∞} . Sometimes we write M r = M r,k for brevity. Note that the norms k K k ( A ) k,r < ∞ fordifferent A > M k,r does not depend on A . T k : ( H k , K k ) ( H k +1 , K k +1 )Here, we introduce the renormalisation step at a scale k , k = 0 , . . . , N − k , the interaction will be split between functions H k and K k . (Hereand in the following we suppress the notation indicating the dependence on q ,reinstating it only when it will play a crucial role.) The “ideal local Hamiltonian”part H k is collecting all relevant (or marginal) directions under the renormalisationtransformation, with all irrelevant ones delegated to the coordinate K k . There isonly limited number of parameters in the relevant coordinate H k . Being given a pair( H k , K k ), H k ∈ M ( B k , X ) and K k ∈ M ( P k , X ), we define a pair ( H k +1 , K k +1 ), H k +1 ∈ M ( B k +1 , X ) and K k +1 ∈ M ( P k +1 , X ), so that(4.46) R k +1 (e − H k ◦ K k )(Λ N , ϕ ) = (e − H k +1 ◦ K k +1 )(Λ N , ϕ )with ( R k +1 F )( X, ϕ ) = R X F ( X, ϕ + ξ ) µ k +1 (d ξ ).As the scale k is fixed in the rest of this chapter, we will skip it and write( H ′ , K ′ ) for ( H k +1 , K k +1 ), with (4.46) becoming(4.47) R (e − H ◦ K ) = e − H ′ ◦ K ′ . To define the Hamiltonian H ′ on the next scale, we first introduce the projection(4.48) Π : M ∗ ( B , X ) → M ( B , X )as a “homogenization” of the second order Taylor expansion T around zero. Namely,for any F ∈ M ∗ ( B , X ) with(4.49) T F ( B, ˙ ϕ ) = F ( B, 0) + DF ( B, ϕ ) + D F ( B, ϕ, ˙ ϕ ) , we define(4.50) Π F ( B, ˙ ϕ ) = F ( B, 0) + ℓ ( ˙ ϕ ) + Q ( ˙ ϕ, ˙ ϕ )so that ℓ is a (unique) linear function of the form (4.19) that agrees with DF ( B, ϕ on ( B ∗ ) ∗ and Q is a (unique) quadratic function ofthe form (4.20) that agrees with D F ( B, 0) on all affine functions ˙ ϕ on ( B ∗ ) ∗ .Strictly speaking, we have in mind functions ˙ ϕ ∈ X such that they are quadratic or affine when restricted to ( B ∗ ) ∗ . Since, for B ∈ B k , k ≤ N − 1, the set ( B ∗ ) ∗ is not wrapped around the torus (as soon as 2 d +2 ≤ L ), we do not need to beconcerned with a possibility of a contradiction in the assumption of ˙ ϕ ∈ X havinga quadratic or affine restriction to ( B ∗ ) ∗ . Clearly, Π F ∈ M ( B , X ) ⊂ M ( B , X )whenever F ∈ M ∗ ( B , X ) and Π F = F for F ∈ M ( B , X ). In particular, we willconsider the projection Π on functions F ∈ M ∗ ( B , X ) of the form(4.51) F ( B, ϕ ) = X X ∈S X ⊃ B | X | F ( X, ϕ )for any F ∈ M ( S , X ).Now we are ready to define the iteration H ′ . Recalling that R = R k +1 is themapping defined by convolution with µ k +1 and starting from H ∈ M ( B , X ) and K ∈ M ( P , X ), we define(4.52) H ′ ( B ′ , ϕ ) = X B ⊂ B ′ Π (cid:0) ( R H )( B, ϕ ) − X X ∈S X ⊃ B | X | ( R K )( X, ϕ ) (cid:1) . To define K ′ , we first replace the original variable H ( B, ϕ ) (or rather H ( B, ϕ + ξ ) in anticipation of the integration R ) by e H ( B, ϕ ), the term in the right hand sidesum above,(4.53) e H ( B, ϕ ) = Π (cid:16) ( R H )( B, ϕ ) − X X ∈S X ⊃ B | X | ( R K )( X, ϕ ) (cid:17) . Writing ˜ I ( B, ϕ ) = exp (cid:8) − e H ( B, ϕ ) (cid:9) instead of the original I ( B, ϕ + ξ ) = exp (cid:8) − H ( B, ϕ + ξ ) (cid:9) , and denoting ˜ J = 1 − ˜ I , we introduce(4.54) e K = ˜ J ◦ ( I − ◦ K. Notice that we are considering here the extension of ˜ I, ˜ J , and I to M ( P , X ), resp. M ( P , X × X ), according to (4.15). Let us stress that the equation above (andin similar circumstances later) is to be interpreted as an algebraic definition validpointwise in the variables ϕ and ξ . It means that e K is actually a function on P × X × X defined explicitly by(4.55) e K ( X, ϕ, ξ ) = X Y,Z ∈P k ( X ) Y ∩ Z = ∅ ˜ J X \ Y ∪ Z ( ϕ ) (cid:0) I ( ϕ + ξ ) − (cid:1) Y K ( Z, ϕ + ξ ) . Occasionally, we are skipping the polymer variable X but wish to keep the fieldvariables and write, slightly misusing the notation, say, e K ( ϕ, ξ ) for the mapping e K ( ϕ, ξ ) : P → R defined by e K ( ϕ, ξ )( X ) = e K ( X, ϕ, ξ ). Then the above algebraicequation reads(4.56) e K ( ϕ, ξ ) = ˜ J ( ϕ ) ◦ (cid:0) I ( ϕ + ξ ) − (cid:1) ◦ K ( ϕ + ξ ) . It is useful to observe that I − ˜ I = ( I − 1) + ˜ J yields I − ˜ I = ˜ J ◦ ( I − 1) and thus e K = ( I − ˜ I ) ◦ K suggesting the interpretation of e K ( ϕ, ξ ) as K ( ϕ + ξ ) combinedwith the perturbation I ( ϕ + ξ ) − ˜ I ( ϕ ). .3. RENORMALISATION TRANSFORMATION T k : ( H k , K k ) ( H k +1 , K k +1 ) 31 Now, using I ( ϕ + ξ ) = ˜ I ( ϕ ) + ˜ J ( ϕ ) + (cid:0) I ( ϕ + ξ ) − (cid:1) , we immediately infer that(4.57) I ( ϕ + ξ ) = ˜ I ( ϕ ) ◦ ˜ J ( ϕ ) ◦ (cid:0) I ( ϕ + ξ ) − (cid:1) and thus(4.58) I ( ϕ + ξ ) ◦ K ( ϕ + ξ ) = ˜ I ( ϕ ) ◦ ˜ J ( ϕ ) ◦ (cid:0) I − (cid:1) ( ϕ + ξ ) ◦ K ( ϕ + ξ ) = ˜ I ( ϕ ) ◦ e K ( ϕ, ξ ) . As a result,(4.59) R ( I ◦ K )(Λ N , ϕ ) = ( ˜ I ◦ ( R e K ))(Λ N , ϕ ) , or, explicitly,(4.60) R ( I ◦ K )(Λ N , ϕ ) = X X ∈P (Λ N ) ˜ I Λ N \ X ( ϕ ) Z X e K ( X, ϕ, ξ ) µ k +1 (d ξ ) . Here we kept the index k + 1 at µ k +1 to avoid a confusion with the measure µ = µ ∗ · · · ∗ µ N +1 .The function K ′ on the next scale satisfying (4.47) will be defined by sorting the X -terms according to the next level closure U . While for any X ∈ P (Λ N ) \ S (Λ N )we attribute the contribution to K ′ ( U ) with U = X ∈ P (Λ N ) ′ , for X ∈ S (Λ N ),we (potentially) split the contribution between several U ’s. Namely, introducingthe factor χ ( X, U ) = |{ B ∈B ( X ): B ∗ = U }|| X | for any X ∈ S (Λ N ) and χ ( X, U ) = 1l U = X for X ∈ P (Λ N ) \ S (Λ N ) (including the case of X consisting of several disjointcomponents from S (Λ N )), we have(4.61) ( ˜ I ◦ e K )(Λ N , ϕ, ξ ) = X U ∈P ′ I ′ Λ N \ U ( ϕ ) h χ ( X, U ) X X ⊂ U ˜ I U \ X ( ϕ ) e K ( X, ϕ, ξ ) i . Here we used the observation that, for any X ∈ S (Λ N ) contributing to several U ’s,we get P U ∈P ′ χ ( X, U ) = 1 and, also, that X ⊂ B ∗ and thus X ⊂ B ∗ .Defining now(4.62) K ′ ( U, ϕ ) = X X ⊂ U χ ( X, U ) ˜ I U \ X ( ϕ ) Z X e K ( X, ϕ, ξ ) µ k +1 (d ξ )for any connected U ∈ P ′ , and extending the definition by taking the correspondingproduct over connected components for a non-connected U , we get(4.63) R ( I ◦ K )(Λ N , ϕ ) = ( I ′ ◦ K ′ )(Λ N , ϕ )in view of (4.60) and (4.61).Notice that if K is L k -periodic, then K ′ is obviously L k +1 -periodic. Also, thetransform conserves the factorisation property of the coordinate K : if K factors onthe scale k ,(4.64) X, Y ∈ P , and X ∩ Y = ∅ , then K ( X ∪ Y, ϕ ) = K ( X, ϕ ) K ( Y, ϕ ) , As will become clear later, the reason for doing so is a need to deal with relevant quadraticterms stemming from K ’s with X ∈ S . In anticipation, those terms are already included as thesecond term in e H ′ (cf. (4.52)) and the particular way of splitting them among U ’s leads to theexact cancelations of the corresponding linearized terms. In particular, the linearization of themap K → K ′ contains only terms starting with the third order in the Taylor expansion of K ( X, ϕ )for X small (cf. (4.83)). Using the fact that only the terms linear in K ( X ) with X ∈ S are relevantin this context, it suffices to introduce a nontrivial χ only for such terms. Our definition is thus aslight simplification of the trick introduced by Brydges [ Bry09 ]. We thank Felix Otto and GeorgMenz for discussions about this point. then K ′ factors on the scale k + 1.Indeed, let X , X ∈ P be such that their closures in P ′ are disjoint. Then(assuming that L > d +2 ) the range L k +1 of the covariance of µ k +1 plus twicethe possible reach of up to 2 d L k of X ∗ and X ∗ out of the closures of X and X ,respectively, does not surpass the minimal distance L k +1 of the closure of X fromthe closure of X , and thus(4.65) ( R e K )( X ∪ X , ϕ ) = ( R e K )( X , ϕ )( R e K )( X , ϕ ) , inheriting the property from K , I , and ˜ I . Now it is easy to observe that thisfact actually means that K ′ factors, as the pairs of sets contributing, accordingto (4.62), to K ′ ( U , ϕ ) and K ′ ( U , ϕ ) with disjoint U and U are necessarily asdiscussed above.Let us summarise, reinstating the index k , what we have got. Proposition . Let k ∈ { , . . . , N − } , H k ∈ M ( B k , X ) , and K k ∈ M ( P k , X ) be such that it factors. Let H k +1 ∈ M ( B k +1 , X ) be defined by (4.66) H k +1 ( B ′ , ϕ ) = X B ∈B k ( B ′ ) e H k ( B, ϕ ) , where (4.67) e H k ( B, ϕ ) = Π (cid:16) ( R k +1 H k )( B, ϕ ) − X X ∈S k X ⊃ B | X | k ( R k +1 K k )( X, ϕ ) (cid:17) . Using e K k ( ϕ, ξ ) = (cid:0) − e − ˜ H k ( ϕ ) (cid:1) ◦ (cid:0) e − H k ( ϕ + ξ ) − (cid:1) ◦ K k ( ϕ + ξ ) , let K k +1 ∈ M ( P k +1 , X ) be defined by (4.68) K k +1 ( U, ϕ ) = X X ∈P k ( U ) χ ( X, U ) exp n − X B ∈B k ( U \ X ) e H k ( B, ϕ ) o Z X e K k ( X, ϕ, ξ ) µ k +1 (d ξ ) for any connected U ∈ P ′ , with (4.69) χ ( X, U ) = ( |{ B ∈B k ( X ): B ∗ = U }|| X | if X ∈ S k (Λ N ) , U = X if X ∈ P k (Λ N ) \ S k (Λ N ) , and by the corresponding product over connected components for any non-connected U . Then K k +1 ∈ M ( P k +1 , X ) , it factors, and (4.70) R k +1 (e − H k ◦ K k )(Λ N , ϕ ) = (e − H k +1 ◦ K k +1 )(Λ N , ϕ ) . As a result, introducing(4.71) T k ( H k , K k , q ) = ( H k +1 , K k +1 )with H k +1 and K k +1 defined by equations (4.66 – 4.68), we get the renormalizationmap(4.72) T k : M ( B k , X ) × M ( P k , X ) × R d × d sym → M ( B k +1 , X ) × M ( P k +1 , X ) ,k = 0 , , . . . , N − .4. KEY PROPERTIES OF THE RENORMALISATION TRANSFORMATION 33 Of course, defining the renormalisation map T k satisfying (4.70) is only halfof our task of the definition of the renormalisation transform. Another part lies inthe verification that the choice of coordinates H k and K k together with the map( H k , K k ) ( H k +1 , K k +1 ) indeed isolates relevant and irrelevant variables withcorrect estimates. Notice that in the definition of T k , we explicitly included thedependence on the matrix q . It stems from the dependence of the starting Gaussianmeasure µ = µ C ( q ) (and of the corresponding generalised Laplacian A ( q ) ) on q and ittransfers into such a dependence also for the operators C ( q ) ( k ) obtained from the finiterange decomposition, for the corresponding Green functions C ( q ) k, and the measures µ k , and, eventually, for the operators T k . Even though this dependence often doesnot appear in our notation, in the following two Propositions, where we state itskey properties, we explicitly address this dependence and make it thus explicit alsoin the notation. For variables H and K we again skip the subscript k and replace k + 1 by a prime.It is easy to verify that, for any q , the origin ( H, K ) = (0 , 0) is a fixed point ofthe transformation T k . Further, the H -coordinate of the operator T k has actuallya linear dependence; we can write(4.73) T k ( H, K, q ) = ( A ( q ) k H + B ( q ) k K, S k ( H, K, q ))with appropriate linear operators A ( q ) k and B ( q ) k . While delegating the discussion ofthe explicit form and the properties of these operators (as well as the linearizationof the map S k ) to Proposition 4.7, we begin with the smoothness of the nonlinearpart S k .The map S k is given as a composition of several maps and its smoothness will bea consequence of the smoothness of the composing maps. To verify its smoothnesswe find it useful to introduce a notion differentiability that is rather easy to verify. Definition . Let X and Y be normed linear spaces and U ⊂ X be open.We use C m ∗ ( U , Y ) to denote the set of functions G : U → Y such that for each j ≤ m and ˙ x ∈ X , the directional derivative(4.74) D j f ( x, ˙ x j ) = d j d t j G ( x + t ˙ x ) (cid:12)(cid:12)(cid:12) t =0 at any x ∈ U exists and the map ( x, ˙ x ) ∈ U × X → D j G ( x, ˙ x j ) ∈ Y is continuous.The technical reasons for this definition will be apparent later and are explainedin great detail in Appendix D. It turns out that this notion is weak only apparently.In particular, for m ≥ C m +1 ∗ ( U , Y ) is contained in the usual space C m ( U , Y ) of Fr´echet differentiable functions (with operator norms on multilinearforms from L m ( X , Y )), see Proposition D.17.Exploring the smoothness of the nonlinear part S k of the operator T k , we runinto problems stemming from a loss of regularity when deriving S k with respect tothe parameter q . For example, it turns out that(4.75) k D j ′ D j ′′ D ℓ S k ( H, K, q )( ˙ H j ′ , ˙ K j ′′ , ˙ q ℓ ) k ( A ) k +1 ,r − ℓ ≤ C k ˙ H k j ′ ( k ˙ K k ( A ) k,r ) j ′′ k ˙ q k ℓ , where the norm k·k ( A ) k +1 ,r − ℓ in the target space is weaker than the norm k·k ( A ) k,r inthe domain space. As a result we are compelled to consider the map S k with asuitable sequence of normed spaces M = M r ֒ → M r − ֒ → . . . ֒ → M r − m , r > m , defined as the spaces M r ( P c k , X ) endowed with the norms k·k ( A ) k,r , r = r , r − , . . . , r − m , respectively, and the space M defined as M ( B k , X ) withthe norm k·k k, . Similarly, M ′ = M ′ r ֒ → M ′ r − ֒ → . . . ֒ → M ′ r − m are definedas M ( P c k +1 , X ) with the norms k·k ( A ) r,k +1 , r = r , r − , . . . , r − m . Further, wewill use f M r to denote the closure of M in M r , and similarly for f M ′ r .Considering now open subsets U ⊂ M × M and V ⊂ R d × d sym , we will introducethe class of functions that can be described as those G : U × V → M ′ for which thederivative D j ′ D j ′′ D ℓ G is a continuous map U ×V × M j ′′ × f M j ′ r × ( R d × d sym ) ℓ → M ′ r − ℓ .More formally, we introduce the set e C m ( U × V , M ′ ) of maps G : U × V → M ′ asfollows (see Definition D.24 in a more general setting): Definition . Let r , m ∈ N , r > m . We define e C m ( U × V , M ′ ) as theset of all maps G : U × V → M ′ such that(a) G ∈ C m ∗ ( U × V , M ′ r − m ).(b) For each 0 ≤ j ′ + j ′′ + ℓ ≤ m , the function( H, K, q , ˙ H , . . . , ˙ H j ′ , ˙ K , . . . , ˙ K j ′′ , ˙ q , . . . , , ˙ q ℓ ) →→ D j ′ D j ′′ D ℓ G (( H, K, q ) , ˙ q , . . . , , ˙ q ℓ , ˙ K , . . . , ˙ K j ′′ , ˙ H , . . . , ˙ H j ′ ) , (which is by an implication of the claim (a) (see Theorem D.10) defined asa map U × V × M j ′ × M j ′′ × ( R d × d sym ) ℓ → M ′ r − m ) has an extension to acontinuous mapping U × V × M j ′ × f M j ′′ r − m +2 ℓ × ( R d × d sym ) ℓ → M ′ r − m . Thisextension is also denoted D j ′ D j ′′ D ℓ G .(c) For each 0 ≤ j ′ + j ′′ + ℓ ≤ m and r = r , r − , . . . , r − m + 2 ℓ , the restrictionof D j ′ D j ′′ D ℓ G to U ×V × M j ′ × f M j ′′ r × ( R d × d sym ) ℓ (notice that it has been alreadyextended by (b)) has values in M ′ r − ℓ and is continuous as a mapping betweenthese spaces.Again, see Appendix D for further context and properties of the notion ofsmoothness introduced in this way. Contrary to Definition D.24 we abstain frominvoking the relevant sequences of normed spaces in the notation as here they arefixed from the context.In the following we will consider the constants d , ω , and r to be fixed (assuming d = 2 , ω ≥ d d +1 + 1) and we will not mention possible dependence of variousconstants (like L , h , and A below) on it. For the proof of the results in Chapter 2 r = 9 is sufficient, see comment in Remark 4.8).For fixed values of the parameters L, h , and A in the definition of the normsin Chapter 4.2, let U ρ ⊂ M × M r and V ⊂ R d × d sym be the neighbourhoods of theorigin,(4.76) U ρ = { ( H, K ) ∈ M × M r : k H k k, < ρ, k K k ( A ) k,r < ρ } and(4.77) V = { q ∈ R d × d sym : k q k < / } . Proposition S k ) . There exists a con-stant L and, for any L ≥ L , constants h ( L ) and A ( L ) , and for any A ≥ A a .4. KEY PROPERTIES OF THE RENORMALISATION TRANSFORMATION 35 constant ρ = ρ ( A ) such that, for any k = 0 , . . . , N − , any L ≥ L , h ≥ h , and A ≥ A we have (4.78) S k ∈ e C m ( U ρ × V , M ′ ) , and there is a constant C = C ( L, h, A ) > such that (4.79) k D j ′ D j ′′ D ℓ S k ( H, K, q )( ˙ H j ′ , ˙ K j ′′ , ˙ q ℓ ) k ( A ) k +1 ,r − ℓ ≤ C k ˙ H k j ′ ( k ˙ K k ( A ) k,r ) j ′′ k ˙ q k ℓ , for any ( H, K ) ∈ U ρ , q ∈ V , ≤ j ′ + j ′′ + ℓ ≤ m , and r = r , r − , . . . , r − m +2 ℓ . The proof will be deferred to Chapter 6, where we will split S k into a com-position of several partial maps and deal with their smoothness separately, iso-lating in detail the needed restrictions on various constants. Here, instead, weoffer a heuristic explanation of the role of the principal constants. The restrictionson L are purely geometric (see Lemma 5.1, Lemma 7.1, Lemma 7.2, Lemma 7.3,Lemma 7.8). In particular, by assuming that L ≥ L we have L ≥ d +1 imply-ing, for example, that if B ∈ B k , then the cube B ∗ has the side at most L k +1 and thus B ∗ ∈ S k +1 . The restrictions on the constant h are more subtle (seeLemma 5.1, Lemma 7.1, Lemma 7.2, Lemma 7.3). Its role is to suppress largefields in the norms k F ( X ) k k,X,r and |k F ( X ) k| k,X by employing the h -dependentweight factors W Xk and w Xk , respectively. When evaluating the norms of the maps( H, K ) → e H (see (4.67)) and K → R k +1 ( K ), a major part of the coarse grainedincrease is absorbed into the growth L k → L k +1 of the corresponding factors inthe functions G k,x and g k,x entering the weight factors. However, some surplusremains stemming essentially from the term L η ( n,d ) in the fluctuation bound (4.3)of the finite range decomposition. A suppression of the relevant term is obtainedby assuming that h ≥ h ( L ) = h L d +5 d +16 with h depending only on d and ω .Finally, the constant A is responsible for combining the norms k·k k,X,r into a singlenorm k·k ( A ) k,r (see Lemma 6.10 and Lemma 7.2). However, it turns out that the map K → R k +1 ( K ) leads to acquiring a factor 2 | X | k in the norm k·k k,X,r , yielding aninevitable loss in A in the norm k·k ( A ) k,r . Nevertheless, the loss can be recovered whencombining the terms in (4.68) while passing to the next scale. Namely, using in theresulting sum stemming from evaluating the norm of (4.68) the geometric bound | X | k ≥ (1 + α ( d )) | X | k +1 − (1 + α ( d ))2 d +1 |C ( X ) | with a constant α ( d ) > 0, weget the original A once we suppose that the map is restricted to sufficiently smalldomain, e.g. assuming that k R k +1 ( K ) k ( A ) k : k +1 ,r ≤ ρ ( A ) = (2 A d +3 ) − and taking A sufficiently large depending on L (and d ).The next claim deals with the linearisation of the map T k at the fixed point( H, K ) = (0 , L between Banach spaces, we considerhere the standard norm k L k = sup {k L ( f ) k : k f k ≤ } , with appropriate normson the corresponding spaces. Usually we indicate the corresponding norms in anappropriate way, e.g., k L k k,r ; k +1 , and k L k k,r ; k +1 ,r , or simply k L k r ;0 and k L k r , fora linear mapping L : M r → M ′ and L : M r → M ′ r , respectively. Proposition T k ) . The first derivative at H = 0 and K = 0 have a triangular form, (4.80) D T k (0 , , q )( ˙ H, ˙ K ) = A ( q ) k B ( q ) k C ( q ) k ! (cid:18) ˙ H ˙ K (cid:19) , with (4.81) ( A ( q ) k ˙ H )( B ′ , ϕ ) = X B ∈B ( B ′ ) (cid:2) ˙ H ( B, ϕ ) + X x ∈ B d X i,j =1 ˙ d i,j ∇ i ∇ ∗ j C ( q ) k +1 (0) (cid:3) , (4.82) ( B ( q ) k ˙ K )( B ′ , ϕ ) = − X B ∈B ( B ′ ) Π X X ∈S X ⊃ B | X | (cid:16)Z X ˙ K ( X, ϕ + ξ ) µ ( q ) k +1 (d ξ ) (cid:17) , and (4.83) ( C ( q ) k ˙ K )( U, ϕ ) = X B : B ∗ = U (cid:0) − Π (cid:1) X Y ∈S Y ⊃ B | Y | (cid:16)Z X ˙ K ( Y, ϕ + ξ ) µ ( q ) k +1 (d ξ ) (cid:17) ++ X X ∈P c \S X = U Z X ˙ K ( X, ϕ + ξ ) µ ( q ) k +1 (d ξ ) . Further, let θ ∈ (1 / , / and let L and h = h ( L ) be as in Proposition 4.6.There exists a constant M = M ( d ) and, for any L ≥ L , a constant A = A ( L ) ,such that for any h ≥ h ( L ) and any A ≥ A ( L ) , the following bounds on thenorms of operators A ( q ) k , B ( q ) k , and C ( q ) k hold independently of N and k and forany k q k ≤ : (4.84) k C ( q ) k k r ≤ θ, k A ( q ) k − k r ; r ≤ √ θ , and k B ( q ) k k r ;0 ≤ M L d ,r ≥ , and for all A ≥ A (note that for the contraction bound for C ( q ) the choice h ≥ h is sufficient). Remark . (i) Notice that as a consequence of Proposition 4.6, the operators A ( q ) k , B ( q ) k , and C ( q ) k are m -times differentiable with respect to q , k q k ≤ , andthere exists a finite constant C = C ( h, L ) > k ∂ ℓ q A ( q ) k ˙ H k ≤ C k ˙ H k , k ∂ ℓ q B ( q ) k ˙ K k ≤ C k ˙ K k ℓ +2 , k ∂ ℓ q C ( q ) k ˙ K k r − ℓ ≤ C k ˙ K k r , for any ℓ = 1 , , . . . , m and any r ≥ ℓ + 3 and A ≥ A .(ii) For the results in Chapter 2 we need m = 3. Thus r = 9 is sufficient. ⋄ Proof of Proposition 4.7. Here, we will only show the validity of the ex-plicit formulas for the operators A ( q ) k , B ( q ) k , and C ( q ) k . The bounds needed for theremaining claims will be proven in Chapter 7.Starting from (4.66) and (4.67), let us expand the linear and quadratic terms in˙ H ( B, ϕ + ξ ) into the sum of the terms depending on ϕ , ξ , and the term proportinalto ˙ Q ( ϕ, ξ ). Observing that the integral with respect to µ k +1 ( ξ ) of the terms linearin ξ vanishes and that Π ( ˙ H ( B, ϕ )) = ˙ H ( B, ϕ ), we get the expression (4.81) for A ( q ) k once we notice that R X ˙ Q ( ξ, ξ ) µ k +1 (d ξ ) = P x ∈ B P di,j =1 ˙ d i,j ∇ i ∇ ∗ j C ( q ) k +1 (0).The formula (4.82) follows directly from the second term on the right hand sideof (4.67). .5. FINE TUNING OF THE INITIAL CONDITIONS 37 When computing C ( q ) k we first observe that only linear terms in e K can con-tribute. Taking ˙ H = 0 and using thus (4.68) with(4.86) e H ( B, ϕ ) = − Π X X ∈S X ⊃ B | X | ( R ˙ K )( X, ϕ )and e K ( ϕ, ξ ) = (cid:0) − e − ˜ H ( ϕ ) (cid:1) ◦ K ( ϕ + ξ ), we get(4.87) C ( q ) k ( ˙ K )( U, ϕ ) = X Y ∈S χ ( Y, U ) Z X D e K (0)( ˙ K )( Y, ϕ, ξ ) µ k +1 (d ξ )++ X X ∈P c \S X = U Z X D e K (0)( ˙ K )( X, ϕ, ξ ) µ k +1 (d ξ ) . Writing χ ( Y, U ) = P B ∈ YB ∗ = U | Y | and observing that(4.88) D e K (0)( ˙ K )( B, ϕ, ξ ) = ˙ K ( B, ϕ + ξ ) − D e − ˜ H (0) ( ˙ K )( B, ϕ ) for Y = B, (4.89) D e K (0)( ˙ K )( Y, ϕ, ξ ) = ˙ K ( Y, ϕ + ξ ) for Y = B, and(4.90) D e − ˜ H (0) ( ˙ K )( B, ϕ ) = Π X Y ∈S Y ⊃ B | Y | (cid:0) R ˙ K (cid:1) ( Y, ϕ ) , we get (4.83). (cid:3) Our next task is to implement in detail the idea of fine tuning outlined inChapter 3. More specifically we will choose an initial ideal Hamiltonian (as used in(3.15) and defined in (4.17)),(4.91) H ( x, ϕ ) = λ + d X i =1 a i ∇ ϕ ( x ) + d X i,j =1 c i,j ∇ i ∇ j ϕ ( x ) + 12 d X i,j =1 q i,j ∇ ϕ ( x ) ∇ j ϕ ( x )such that the final ideal Hamiltonian vanishes (note that in Chapter 3 weconsidered only the simplified case λ = a = c = 0).Given an initial K we want to evaluate the integral Z N ( u ) = Z X N Y x ∈ Λ (cid:0) K ( x, ϕ ) (cid:1) µ (d ϕ ) = Z X N (1 ◦ K )(Λ , ϕ ) µ (d ϕ ) . Analogously to the calculation in Chapter 3 cf. (3.16) we can rewrite this integralas(4.92) Z N ( u ) = Z X N e H (Λ ,ϕ ) (cid:0) e −H ◦ e −H K (cid:1) (Λ , ϕ ) µ (d ϕ )= Z ( q ) N Z (0) N e L dN λ Z X N (cid:0) e −H ◦ e −H K (cid:1) (Λ , ϕ ) µ ( q ) (d ϕ )where Z ( q ) N and Z (0) N are as in Chapter 3. Here we used that P x ∈ Λ ∇ i ϕ ( x ) = 0 and P x ∈ Λ ∇ i ∇ j ϕ ( x ) = 0 because ϕ is periodic. We will now show that for sufficiently small K there exists an H = H ( K ) suchthat the second integral in (4.92) deviates from 1 only by an exponential small termand such that the derivatives of this term with respect to K are also controlled.To do so we proceed in two steps. We first show that given sufficiently small K and H there exists an ideal Hamiltonian F ( K , H ) ∈ M and a small ’irrelevant’term F N ( K , H ) ∈ M N,r such that(4.93) Z X N (cid:0) e −F ( K , H ) ◦ e −H K (cid:1) (Λ , ϕ ) µ ( q ) (d ϕ ) = Z X N (1 + F N ( K , H )) µ ( q ) N +1 (d ϕ ) . As a byproduct of this construction we will see that for K = 0 we have F (0 , H ) = 0and F N (0 , H ) = 0 for all sufficiently small H . Together with smoothness resultsfor F this implies D H F (0 , 0) = 0 and the implicit function will guarantee thatthere exists a unique map H mapping a neighbourhood of the origin in E to M such that(4.94) F ( K , H ( K )) = H ( K ) . Combining this with (4.93) and (4.92) we get(4.95) − log Z N ( u ) = − log Z ( q ) N Z (0) N − λL dN − log Z X N (1 + F N ( K , H ( K ))) µ ( q ) N +1 (d ϕ ) , where(4.96) λ = π ( H ( K )) and q = π ( H ( K ))denote the constant term in H ( K ) and the coefficient matrix of the quadratic term,respectively.We now first explain how to construct the maps F and F N . We rewrite theentire cascade of maps T k in terms of a single map on a suitably defined Banachspace. First, we introduce the Banach spaces(4.97) Y r = (cid:8) y = ( H , H , K , . . . , H N − , K N − , K N ) : H k ∈ M k, , K k ∈ M k,r (cid:9) with the norms(4.98) k y k Y r = max k ∈{ ,...,N − } η k k H k k k, ∨ max k ∈{ ,...,N } αη k k K k k k,r for r = 1 , . . . , r and with parameters η ∈ (0 , 1) and α ≥ k also in the notation for normed spaces;we write M k, and M k,r instead of M and M r used previously. Notice that theterms K and H N are not present in y ∈ Y r ; while the latter is put to be 0, theformer is singled out as an initial condition for a separate treatment. Also, noticethat k y k Y r ≤ k y k Y r +1 and thus Y r +1 ֒ → Y r .Taking into account the dependence of T k on q (the matrix in the quadraticterm of H ) and on the initial perturbation K ∈ E (see (2.21)) we define the map(4.99) T : Y r × E × M → Y r by(4.100) T ( y , K , H ) = y . Here, y is given by recursive equations,(4.101) H k = A − k (cid:0) H k +1 − B k K k (cid:1) ,K k +1 = S k ( H k , K k , q ) = C k K k + S k ( H k , K k , q ) − D S k ((0 , , q ) , K k ) . .6. PROOF OF STRICT CONVEXITY—THEOREM 2.1 39 for k = 0 , . . . , N − 1. Here C k K k = D S k ((0 , , q ) , K k ) and S k ( H k , K k , q ) − D S k ((0 , , q ) , K k ) is the nonlinear part of the map S k . In addition, we set H N = 0and define K ∈ M ( P , X ) by K = e −H K , i.e., by(4.102) K ( X, ϕ ) := Y x ∈ X (cid:0) exp( −H ( x, ϕ )) K ( ∇ ϕ ( x )) (cid:1) with K ∈ E and H ∈ M .Observe now that, for a given K and H , the 2 N -tuple y is a fixed point of T ,i.e., T ( y , K , H ) = y if and only if(4.103) T k ( H k , K k , q ) = ( H k +1 , K k +1 ) , k = 0 , . . . , N − , with K = e −H K and H N = 0. Our task thus is to find a map F from a neigh-bourhood of origin in E × M to Y r so that(4.104) T ( F ( K , H ) , K , H ) = F ( K , H ) . This can be done with help of the Implicit Function Theorem E.1 using thebounds from Propositions 4.7 and 4.6 to verify its hypothesis. In Proposition 8.1,we will summarize the smoothness properties of the obtained fixed point map F .Note that for K = 0 the vector y = 0 is a fixed point for every H . Thus(4.105) F (0 , H ) = 0 . Taking now for F and F N the first and last component of F , correspondingto H and K N , the equality (4.93) readily follows from the definition of F .Now we can easily construct the map H . The condition (4.105) and the differ-entiability of F (see Proposition 8.1) imply that(4.106) D H F (0 , 0) = 0 . Thus we can apply the implicit function theorem in the space C m ∗ to get the fol-lowing result. Theorem . Let m + 3 ≤ r . There exist constants ρ , ρ > , and aparameter ζ > in the definition of the norm on the space E introduced in (2.21) such that there exists a C m ∗ -map H : B E ( ρ ) → B M ( ρ ) satisfying the fixed pointequations (4.107) F ( K , H ( K ))) = H ( K ) and (4.108) T ( F ( K , H ( K )) , K , H ( K )) = F ( K , H ( K )) for all K ∈ B E ( ρ ) . Moreover, the C m ∗ - norm of the map H is bounded uniformlyin N . We may choose ρ < h . Then in view of (4.44) the matrix q = π ◦ H ( K ) of the quadratic part of H ( K ) satisfies | q | < . We are following the strategy outlined in Chapter 3, but we now consider thefull ideal Hamiltonian H in (4.91) and not just the quadratic part. To prove thestrict convexity of the surface tension σ β ( u ), we need to prove that its perturbativecomponent ς ( u ) is smooth in the tilt u . This amounts to obtaining a uniform bound(in N ∈ N ) on the approximation(4.109) ς N ( u ) := − L dN log Z N ( u ) with Z N ( u ) defined in (2.17). In view of the equality (4.95), applied with K = K u ,we have(4.110) ς N ( u ) = − L dN log (cid:16) Z ( q ) N Z (0) N (cid:17) − λ + 1 L dN log (cid:16) Z X N (cid:16) F N ( K u , H ( K u ))(Λ N , ϕ ) (cid:17) µ ( q ) N +1 (d ϕ ) (cid:17) , where, as in (4.96),(4.111) λ = π ( H ( K u )) and q = π ( H ( K u ))denote the constant term in H ( K u ) and the coefficient matrix of the qudratic term,respectively.The proof of strict convexity thus consists of the following three steps. Step 1: Choose all needed constants according to Propositions 4.6 and 4.7. Inparticular, we choose (with a fixed d ) the constants L , h , A , ¯ ρ = ¯ ρ ( A ), and aconstant C , so that the claims from Propositions 4.6 and 4.7 (i.e., differentiabilityand uniform smoothness of the renormalization maps T k as well as the contractivityof the linearisation) are valid for any ( H, K, q ) ∈ U ρ (in particular, k q k ≤ ). Step 2: Apply Theorem 4.9 to get the existence and smoothness properties of themap H : B E ( ρ ) → B M ( ρ ). Step 3: Finally, address the dependence of K u on the tilt u : according to theassumptions of Theorem 2.1 we have a C tilt map τ , u τ ( u ) = K u . Choosing δ sufficiently small, we have τ ( B δ (0)) ⊂ B E ( ρ ) ⊂ E .Having this in mind, we show that the right hand side of (4.110) is three timescontinuously differentiable in u with bounded derivatives, by analysing each of thethree terms separately.The first term on the right hand side of (4.110) can easily be computed as(4.112) − log (cid:16) Z ( q ) N Z (0) N (cid:17) = log det (cid:0) A ( q ) C (0) (cid:1) . Consider the dual torus(4.113) b T N = n p = ( p , . . . , p d ) : p i ∈ (cid:8) − ( L N − πL N , − ( L N − πL N , . . . , ( L N − πL N (cid:9) , i = 1 , . . . , d o , and the functions f p ( x ) = e i h p,x i . The family (cid:8) | Λ N | − / f p (cid:9) p ∈ b T N \{ } is an orthonor-mal basis of V N . The eigenvalues of A ( q ) are(4.114) σ ( p ) = h q ( p ) , (1l + q ) q ( p ) i = d X l,j =1 q ( p ) l (cid:0) δ l,j + q l,j (cid:1) q ( p ) j , p ∈ b T N with q ( p ) j = e ip j − j = 1 , . . . , d . Note that q ( p ) l q ( p ) j ≈ p l p j . The eigenvalues for A (0) and C (0) are h q ( p ) , q ( p ) i ≈ k p k and h q ( p ) , q ( p ) i − ≈ k p k − , p ∈ b T N , respectively. Weget(4.115) log det (cid:0) A ( q ) C (0) (cid:1) = Tr log (cid:0) A ( q ) C (0) (cid:1) = X p ∈ b T N \{ } log (cid:16) h q ( p ) , q q ( p ) ih q ( p ) , q ( p ) i (cid:17) . .6. PROOF OF STRICT CONVEXITY—THEOREM 2.1 41 Since the sum over the torus has L dN − − L dN log (cid:16) Z ( q ) N Z (0) N (cid:17) is a smooth function of q with derivatives bounded uniformly in N . Thus u 7→ − L dN log (cid:16) Z π ( H ( K u )) N Z (0) N (cid:17) is a C ∗ mapping with uniformly bounded derivatives. Note that the chain ruleinitially states that this map is C ∗ , but R d being a finite dimensional vector spaceit is actually a C mapping according to Proposition D.17.As regards the second term we know from Theorem 4.9 and the chain rule that u H ( K u ) is C ∗ . Thus the map u λ = π ( H ( K u )) is C ∗ and hence C becausethe map is defined a neighbourhood in the finite dimensional space R d .Regarding the last termlog (cid:16) Z X N (cid:16) F N ( K u , H ( K u ))(Λ N , ϕ ) (cid:17) µ ( q ) N +1 (d ϕ ) (cid:17) we first note that for a positive function G the k -th derivative of log G is a polyno-mial in G and the first k derivatives of G . Since µ ( q ) N +1 is a probability measure, itsuffices to show that(4.116) (cid:12)(cid:12)(cid:12)(cid:12)Z X N F N ( K u , H ( K u ))(Λ N , ϕ ) µ ( q ) N +1 (d ϕ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ . and to estimate the derivatives of the integral. We thus need to estimate(4.117) T ( u ) := Z X N F N ( K u , H ( K u ))(Λ N , ϕ ) µ ( q ) N +1 (d ϕ ) , where q = π ( H ( K u )),and its derivatives with respect to u . The integral in (4.117) is exactly the appli-cation of the renormalisation map R , defined in (6.16), evaluated at zero: T ( u ) = ( R ( q )1 P )(Λ N , 0) where P = F N ( K u , H ( K u )) and q = π ( H ( K u )).Thus we can apply the estimates for R stated in Lemma 6.5 and in Lemma 5.1(iv). We introduce the notation e R ( K, H ) := ( R ( q )1 K )(Λ N , 0) = R ( K, q )(Λ N , . It will later be convenient to view ˜ R as a function of K and H even thus it dependson H only through q = π ( H ). We get T ( u ) = e R (cid:16) F N ( K u , H ( K u )) , H ( K u ) (cid:17) . Now by Lemma 5.1 (iv) (note that there is only one N -block), Proposition 8.1, thedefinition (4.98) of the norm on F , Theorem 4.9 and the assumptions on K u inTheorem 2.1 we get | T ( u ) | ≤ kF N ( K u , H ( K u )) k ≤ η N α kF ( K u , H ( K u )) k Y ≤ C η N α . Thus (4.116) holds if N is large enough (note that α and C are independent of N ).To verify the differentiability of T we recall the notation( F ⋄ G )( x, H ) = F ( G ( x, H ) , H ) to rewrite T ( u ) as T ( u ) = (cid:0) e R ⋄ F N (cid:1) ( K u , H ( K u ))Now by Proposition 8.1 we have F N ∈ e C m ( B X × M ( b ρ , b ρ ) , Y ) with boundson the derivatives which are independent of N . Here Y = Y r ֒ → Y r − ֒ → . . . ֒ → Y r − m and in the domain we use the trivial scale X m = . . . = X = E .By Lemma 6.5 we have e R ∈ e C m ( Y × B b ρ , R ) (as long as b ρ < h ), again withbounds on the derivatives which are independent of N . Thus the chain rule withloss of regularity, Theorem D.29, shows that e R ⋄ F N ∈ e C m ( B X × M ( b ρ , b ρ ) , R )with uniformly bounded derivatives. Since the scale X m = . . . = X = E is trivial(and since the target is just R ) this implies that e R ⋄ F N ∈ C m ∗ ( B X × M ( b ρ , b ρ ) , R )Together with the regularity of H (see Theorem 4.9) and the assumptions on K u in Theorem 2.1 we get T ∈ C ∗ ( B ( δ )) with uniformly bounded derivatives. Since B ( δ ) ⊂ R d by Proposition D.17 this is the same as T ∈ C ( B ( δ )). (cid:3) HAPTER 5 Properties of the Norms As a preparation for the proof of Propositions 4.7 and 4.6, we first address thefactorisation properties of the norms defined in Chapter 4.2 and prove a bound onthe integration map R k defined in (4.11). Recalling that the norms k · k k,X,r dependon parameters L, h , and ω , we summarise their properties in the following lemma.Using η ( n, d ) defined by (4.4), we introduce κ ( d ) := (cid:0) d + η (2 ⌊ d +22 ⌋ + 8 , d ) (cid:1) with ⌊ t ⌋ denoting the integer value of t . Notice that κ ( d ) ≤ d / d + 16. Lemma . Let ω ≥ √ , N ∈ N , N ≥ , and L ∈ N odd, L ≥ . Given k ∈ { , . . . , N − } , let K ∈ M ( P k , X ) factor (at the scale k ), andlet F ∈ M ( B k , X ) . Then, the norms k·k k,X,r , k·k k : k +1 ,X,r , r ∈ { , . . . , r } , and |k·k| k,X , X ∈ P k , satisfy the following conditions:(i) k K ( X ) k k,X,r ≤ Q Y ∈C ( X ) k K ( Y ) k k,Y,r and k K ( X ) k k : k +1 ,X,r ≤ Q Y ∈C ( X ) k K ( Y ) k k : k +1 ,Y,r ,(iia) k F X K ( Y ) k k,X ∪ Y,r ≤ k K ( Y ) k k,Y,r |k F k| | X | k k as well as(iib) k F X K ( Y ) k k : k +1 ,X ∪ Y,r ≤ k K ( Y ) k k : k +1 ,Y,r |k F k| | X | k k for X, Y ∈ P k dis-joint,(iii) |k B ) k| k,B = 1 for B ∈ B k ,(iv) There exists a constant h = h ( d, ω ) depending only on the dimension d and value of the parameter ω , such that for any h ≥ L κ ( d ) h and X ∈ P k ,we have k ( R k +1 K )( X ) k k : k +1 ,X,r ≤ | X | k k K ( X ) k k,X,r . Proof. (i) Notice first that for any F , F ∈ M ( P k , X ) and any (not necessarily disjoint) X , X ∈ P k , we have(5.1) | F ( X )( ϕ ) F ( X )( ϕ ) | k,X ∪ X ,r ≤ | F ( X )( ϕ ) | k,X ,r | F ( X )( ϕ ) | k,X ,r . Indeed, using the definition of the norm | · | k,X,r and fact that a Taylor expansionof a product is the product of Taylor expansions, we have(5.2) | F ( X )( ϕ ) F ( X )( ϕ ) | k,X ∪ X ,r ≤ | F ( X )( ϕ ) | k,X ∪ X ,r | F ( X )( ϕ ) | k,X ∪ X ,r . Observing now that for any ˙ ϕ ∈ X N we have | ˙ ϕ | k,X ≤ | ˙ ϕ | k,X ∪ X , we get(5.3) sup | ˙ ϕ | k,X ∪ X ≤ | D s F ( X )( ϕ )( ˙ ϕ, . . . , ˙ ϕ ) | ≤ sup | ˙ ϕ | k,X ≤ | D s F ( X )( ϕ )( ˙ ϕ, . . . , ˙ ϕ ) | , implying(5.4) | F ( X )( ϕ ) | k,X ∪ X ,r ≤ | F ( X )( ϕ ) | k,X ,r and similarly for F , yielding thus (5.1). 434 5. PROPERTIES OF THE NORMS Iterating (5.1) we can use it for K ( X, ϕ ) = Q Y ∈C ( X ) K ( Y )( ϕ ), yielding(5.5) | K ( X, ϕ ) | k,X,r ≤ Y Y ∈C ( X ) | K ( Y )( ϕ ) | k,Y,r and, similarly,(5.6) | K ( X, ϕ ) | k +1 ,X,r ≤ Y Y ∈C ( X ) | K ( Y )( ϕ ) | k +1 ,Y,r To conclude, it then suffices to observe that(5.7) w Xk ( ϕ ) = Y Y ∈C ( X ) w Yk ( ϕ ) and w Xk : k +1 ( ϕ ) = Y Y ∈C ( X ) w Yk : k +1 ( ϕ ) . Here, in both cases, we use the fact that the partition X = ∪ Y ∈C ( X ) Y splits both X and its boundary ∂X into disjoint components: Y , Y ∈ C ( X ), Y = Y impliesthat dist( Y , Y ) > L k and thus Y ∩ Y = ∅ , ∂Y ∩ ∂Y = ∅ , and ∂X = ∪ Y ∈C ( X ) ∂Y .(iia) Using (iterated) (5.1) for Q B ∈B k ( X ) F ( B )( ϕ ) K ( Y )( ϕ ) , we have(5.8) | (cid:0) F X K ( Y ) (cid:1) ( ϕ ) | k,X ∪ Y,r ≤ Y B ∈B k ( X ) | F ( B )( ϕ ) | k,B,r | K ( Y )( ϕ ) | k,Y,r . Bounding the right hand side by(5.9) Y B ∈B k ( X ) |k F ( B ) k| k,B k K ( Y ) k k,Y,r Y B ∈B k ( X ) W Bk ( ϕ ) w Yk ( ϕ ) , we get (ii) once we verify that(5.10) Y B ∈B k ( X ) W Bk ( ϕ ) w Yk ( ϕ ) ≤ w X ∪ Yk ( ϕ ) . Inserting the definitions of the strong and weak weight functions, (5.10) is satisfiedonce(5.11) L k X x ∈ ∂Y G k,x ( ϕ ) ≤ X x ∈ X (cid:0) d ωg k,x ( ϕ ) + ( ω − G k,x ( ϕ ) (cid:1) + L k X x ∈ ∂ ( X ∪ Y ) G k,x ( ϕ ) . To verify this, it suffices to notice that each y ∈ ∂Y \ ∂ ( X ∪ Y ) is necessarilycontained in ∂B for some B ∈ B k ( X ) (a block on the boundary of X touching Y ).Thus, it suffices to show that for each such B one has(5.12) L k X x ∈ ∂B G k,x ( ϕ ) ≤ X x ∈ B (cid:0) d ωg k,x ( ϕ ) + ( ω − G k,x ( ϕ ) (cid:1) . Indeed, applying Proposition B.5 (a), we have(5.13) h L k X x ∈ ∂B G k,x ( ϕ ) ≤≤ c (cid:0)X x ∈ B |∇ ϕ ( x ) | + L k X x ∈ U ( B ) |∇ ϕ ( x ) | (cid:1) + L k X x ∈ ∂B X s =2 L (2 s − k |∇ s ϕ ( x ) | ≤≤ h c X x ∈ B G k,x ( ϕ ) + h c L k X z ∈ ∂B g k,z ( ϕ ) , . PROPERTIES OF THE NORMS 45 where z is any point z ∈ B . Observing that the size of the set ∂B is at most( L k + 2) d − ( L k − d ≤ d L ( d − k once 2 ≤ L , we get the seeked bound once(5.14) 2 c ≤ ω − . Observing that c < √ 2, this condition is satisfied with our choice of ω .(iib) The proof is similar, with (5.11) replaced by(5.15) 3 L k X x ∈ ∂Y G k,x ( ϕ ) ≤ X x ∈ X (cid:0) (2 d ω − g k : k +1 ,x ( ϕ ) + ( ω − G k,x ( ϕ ) (cid:1) + 3 L k X x ∈ ∂ ( X ∪ Y ) G k,x ( ϕ )that, in its turn, needs (5.12) in a slightly stronger version,(5.16) 3 L k X x ∈ ∂B G k,x ( ϕ ) ≤ X x ∈ B (cid:0) (2 d ω − g k : k +1 ,x ( ϕ ) + ( ω − G k,x ( ϕ ) (cid:1) . This is satisfied once(5.17) 6 c ≤ ω − . (iii) follows immediately from the definition.(iv) Since convolution commutes with differentiation we have(5.18) D s Z K ( ϕ + ξ ) µ k +1 (d ξ ) = Z D s K ( ϕ + ξ ) µ k +1 (d ξ ) . For a vector ( A , A , . . . , A r ) consisting of A ∈ R and multilinear symmetric maps A s : X ⊗ s → R , s ∈ N , we consider the norm(5.19) | ( A , . . . , A r ) | := r X s =0 s ! | A s | k +1 ,X with | A s | k +1 ,X defined by (4.24). Then | K ( ϕ ) , DK ( ϕ ) , . . . , D r K ( ϕ )) | = | K ( ϕ ) | k +1 ,X,r . Now fix ϕ and apply Jensen’s inequality to map ξ ( K ( ϕ + ξ ) , . . . , D r K ( ϕ + ξ )).This yields(5.20) (cid:12)(cid:12)(cid:12) Z K ( ϕ + ξ ) µ k +1 (d ξ ) (cid:12)(cid:12)(cid:12) k +1 ,X,r = Z | K ( ϕ + ξ ) | k +1 ,X,r µ k +1 (d ξ ) . Since(5.21) | ˙ ϕ | k,X ≤ L − d | ˙ ϕ | k +1 ,X , we also have(5.22) | K ( X, ϕ + ξ ) | k +1 ,X,r ≤ | K ( X, ϕ + ξ ) | k,X . As a result,(5.23) k ( R k +1 K )( X ) k k : k +1 ,X,r ≤ sup ϕ Z | K ( X, ϕ + ξ ) | k,X,r µ k +1 (d ξ ) w − Xk : k +1 ( ϕ ) . Estimating the integrand | K ( X, ϕ + ξ ) | k,X,r from above by k K ( X ) k k,X,r w Xk ( ϕ + ξ ) , the proof of the needed bound amounts to showing that(5.24) Z X N w Xk ( ϕ + ξ ) µ k +1 (d ξ ) ≤ | X | w Xk : k +1 ( ϕ ) . As this result will be used also later in different circumstances, we state it as aseparate Lemma. Lemma . Let ω ≥ √ . There exists a constant h = h ( d, ω ) such thatfor any N ≥ , L odd, L ≥ , h ≥ L κ ( d ) h , k ∈ { , . . . , N − } , K ∈ M ( P k , X ) ,and any X ∈ P k , we have (5.25) Z X N w Xk ( ϕ + ξ ) µ k +1 (d ξ ) ≤ | X | k w Xk : k +1 ( ϕ ) . Proof. We will prove the bound (5.25) in three steps: Step 1. Expanding the terms ( ∇ ϕ ( x ) + ∇ ξ ( x )) in P x ∈ X G k,x ( ϕ + ξ ) and usingthe Cauchy’s inequality ( a + b ) ≤ a + 2 b for the remaining terms (those thatare preceded by a power in L that allows to absorb the resulting prefactors whilepassing to the next scale), we have(5.26) h X x ∈ X G k,x ( ϕ + ξ ) ≤ X x ∈ X (cid:0) |∇ ϕ ( x ) | + |∇ ξ ( x ) | (cid:1) + 2 (cid:12)(cid:12) X x ∈ X ∇ ϕ ( x ) ∇ ξ ( x ) (cid:12)(cid:12) ++ 2 X x ∈ X (cid:16) L k |∇ ϕ ( x ) | + L k |∇ ξ ( x ) | + L k |∇ ϕ ( x ) | + L k |∇ ξ ( x ) | (cid:17) . For the remaining terms occurring in w Xk ( ϕ + ξ ), we simply write (again by Cauchy’sinequality)(5.27) g k,x ( ϕ + ξ ) ≤ g k,x ( ϕ ) + 2 g k,x ( ξ )and(5.28) L k G k,x ( ϕ + ξ ) ≤ L k G k,x ( ϕ ) + 2 L k G k,x ( ξ ) . Step 2. In view of Proposition B.6, we bound the mixed term 2 (cid:12)(cid:12)P x ∈ X ∇ ϕ ( x ) ∇ ξ ( x ) (cid:12)(cid:12) by(5.29) L k X x ∈ X ∪ ∂ − X |∇ ϕ ( x ) | + L k X x ∈ ∂ − X |∇ ϕ ( x ) | + 1 + c dL k X x ∈ X ∪ ∂ − X ξ ( x ) + c X x ∈ X |∇ ξ ( x ) | . The sum over X in the first term above will be estimated by the regulator g k : k +1 ,x ( ϕ )of the next generation. Namely, combining, for any x ∈ X , its terms with the cor-responding ϕ -terms on the second line in (5.26), we have(5.30) 3 L k |∇ ϕ ( x ) | + 2 L k |∇ ϕ ( x ) | ≤≤ L − L k +1) |∇ ϕ ( x ) | + 2 L − L k +1) |∇ ϕ ( x ) | ≤ L − h g k : k +1 ,x ( ϕ ) , where we are assuming that(5.31) 2 L − ≤ . . PROPERTIES OF THE NORMS 47 The remaining sum over ∂ − X \ X , together with the second term in (5.29), will beabsorbed into the sum P x ∈ ∂X G k,x ( ϕ ). Collecting now all the ϕ -terms in log w k ( ϕ + ξ ) with expanded mixed term, we get the bound(5.32) X x ∈ X d +1 ωg k,x ( ϕ ) + X x ∈ X ωG k,x ( ϕ ) + 3 ωL − X x ∈ X g k : k +1 ,x ( ϕ ) + 3 L k X x ∈ ∂X G k,x ( ϕ ) . This is bounded by(5.33) log w Xk : k +1 ( ϕ ) = X x ∈ X (cid:0) (2 d ω − g k : k +1 ,x ( ϕ ) + ωG k,x ( ϕ ) (cid:1) + 3 L k X x ∈ ∂X G k,x ( ϕ )once(5.34) (3 + 2 d +1 ) ω ≤ (2 d ω − L . This condition, including also (5.31), are satisfied once L ≥ ξ -terms in h log w k ( ϕ + ξ ) with expanded mixed term, weget the bound(5.35) X x ∈ X h d +1 ωg k,x ( ξ ) + X x ∈ X ω (cid:0) (1 + c ) |∇ ξ ( x ) | + 2 L k |∇ ξ ( x ) | + 2 L k |∇ ξ ( x ) | (cid:1) ++ ω (1 + c d ) L − k X x ∈ X ∪ ∂ − X ξ ( x ) + 2 L k X x ∈ ∂X h G k,x ( ξ ) . Bounding the last term with the help of Proposition B.5, we get(5.36) X x ∈ X h d +1 ωg k,x ( ξ ) + X x ∈ U ( X ) (cid:0) ω (1 + c d ) L − k ξ ( x ) + ( ω (1 + c ) + 4 c ) |∇ ξ ( x ) | ++ (2 ω + 8 c ) L k |∇ ξ ( x ) | + (2 ω + 8 c ) L k |∇ ξ ( x ) | + 4 c L k |∇ ξ ( x ) | (cid:1) . Finally, the term g k,x ( ξ ) containing l ∞ -norm of ∇ s ξ , s = 2 , , 4, is boundedwith the help of the Sobolev inequality from Proposition A.1. Taking B ∗ for the B n with n = (2 d +1 − L k , we get(5.37) k∇ s ξ k l ∞ ( B ∗ ) ≤ C (2 d +1 − L kd f M X l =0 L lk X x ∈ B ∗ |∇ l ∇ s ξ | ( x ) , where f M = ⌊ d +22 ⌋ is the integer value of d +22 and in computing the pre-factor wetook into account that 2 ⌊ d +22 ⌋ − d ≤ 2. Notice that the constant C depends (alsothrough f M ) only on the dimension d . As a result, we are getting(5.38) X x ∈ X h d +1 ωg k,x ( ξ ) ≤≤ d +1 ω X x ∈ X X s =2 L (2 s − k C (2 d +1 − L kd M X l =0 L lk X y ∈ B ∗ x |∇ l ∇ s ξ | ( x ) ≤≤ d +1 ω d +1 C (2 d +1 − d +2 L − k M +4 X l =2 L lk X y ∈ X ∗ |∇ l ξ | ( x ) , where in the last inequality we took into account that each point y ∈ X ∗ may accurin B ∗ x for at most (2 d +1 − d L dk points x ∈ X .Summarising, under the conditions (5.31), (5.34), we have(5.39) w Xk ( ϕ + ξ ) ≤ w Xk : k +1 ( ϕ ) exp (cid:16) h − CL k X x ∈ X ∗ M +4 X l =0 L lk |∇ l ξ ( x ) | (cid:17) with the constant(5.40) C = max { ω (1 + c d ) , ω (1 + c ) + 4 c , ω + 8 c ) + 32 d +1 ω C (2 d +1 − d +2 } that depends, afters ω is chosen, only on the dimension d . Step 3. We first bound the term in ξ in (5.39) by a smooth Gaussian and thenbound the remaining integral. Let η X ∗ be a smooth cut-off function such thatsupp η X ∗ ⊂ ( X ∗ ) ∗ , η X ∗ = 1 on X ∗ , and(5.41) (cid:12)(cid:12) ∇ l η X ∗ (cid:12)(cid:12) ≤ ΘL − lk . Then the bound in (5.39) implies taht(5.42) w Xk ( ϕ + ξ ) ≤ w Xk : k +1 ( ϕ ) exp (cid:0) κ ( B k ξ, ξ ) (cid:1) , where κ = 2 Ch − and(5.43) ( B k ξ, ξ ) = 1 L k X x ∈ Λ N M +4 X l =0 L lk (cid:12)(cid:12) η X ∗ ( x )( ∇ l ξ )( x ) (cid:12)(cid:12) . Explicitly,(5.44) B k = B (0) k + M +4 X l =1 B ( l ) k with(5.45) B ( l ) k ξ = 1 L k ( ∇ l ) ∗ η X ∗ ∇ l ξ, l = 1 , . . . , f M + 4 , and B (0) k ξ = 1 L k Π ( η X ∗ ξ ) , where Π : V N → X N is the projection ( Πϕ )( x ) = ϕ ( x ) − | Λ N | P y ∈ Λ N ϕ ( y ) (for l ≥ , ∇ ∗ i ϕ ) = ( ∇ i , ϕ ) = 0).It remains only to show that Z X N exp (cid:0) κ ( B k ξ, ξ ) (cid:1) µ k +1 (d ξ ) ≤ | X | . A formal Gaussian calculation with respect to the measure µ k +1 with the covarianceoperator C k +1 yields(5.46) Z X N exp (cid:0) κ ( B k ξ, ξ ) (cid:1) µ k +1 (d ξ ) = (cid:16) det( C − k +1 − κ B k )det( C − k +1 ) (cid:17) − = det (cid:16) I − κ C k +1 B k C k +1 (cid:17) − . To justify this calculation we will derive a bound on the spectrum σ ( C k +1 B k C k +1 )in the following lemma. Lemma . Using the shorthand η ( d ) := η (2 ⌊ d +22 ⌋ + 8 , d ) = 2 κ ( d ) − d , wehave: . PROPERTIES OF THE NORMS 49 (i) The operators C k +1 B k C k +1 are symmetric and positive definite.There exist constants M and M that depend only on the dimension d such thatfor any N and any k = 1 , . . . , N, (ii) sup σ ( C k +1 B k C k +1 ) ≤ M L d + η ( d ) and(iii) Tr (cid:16) C k +1 B k C k +1 (cid:17) ≤ M | X | k L η ( d ) . Postponing momentarily the proof of the Lemma, we first observe that κ < M L d + η ( d ) with h ≥ L κ ( d ) CM , and thus the eigenvalues λ j , j = 1 , . . . , L Nd − κ C k +1 B k C k +1 lie between 0 and . The formal Gaussian calculation is thenjustified and(5.47)log det (cid:16) I − κ C k +1 B k C k +1 (cid:17) ≥ X i log(1 − λ i ) ≥ X i − λ i = − (cid:16) κ C k +1 B k C k +1 (cid:17) ≥ − M L η ( d ) κ | X | k = − CM L η ( d ) h − | X | k . Hence(5.48) det (cid:16) I − κ C k +1 B k C k +1 (cid:17) − ≤ e CM | X | kh L η ( d ) ≤ e CM | X | kh L − d and the Lemma 5.2 follows with(5.49) h ( d, ω ) ≥ C max (cid:0) M , M d (cid:1) . (cid:3) Proof of Lemma 5.3. The claim (i) follows from definitions.The estimate (ii) follows from the estimate(5.50) k B k C k +1 ξ k ≤ M L d + η ( d ) k ξ k for all ξ ∈ X N . For B (0) k , we first observe that(5.51) L k k B (0) k ξ k = k Π ( η X ∗ ) ξ k ≤ k ( η X ∗ ) ξ k ≤ k ξ k . In view of Proposition 4.1, the operator C k +1 acts by convolution with respect to thefunction C k +1 . With the bounds (5.51), (4.2), (4.3), and c max = max | α |≤ M +4) c α , ,we have (recall that η (0 , d ) ≤ η (2 ⌊ d +22 ⌋ + 8 , d ) = η ( d )))(5.52) k B (0) k C k +1 ξ k ≤ L − k k C k +1 ξ k ≤ L − k X z ∈ Λ N |C k +1 ( z ) |k ξ k ≤ c max L d + η ( d ) k ξ k . For B ( l ) k we use the discrete product rule(5.53) ∇ i ( f g ) = ∇ i f S i g + S i f ∇ i g, where(5.54) ( S i f )( x ) := f ( x ) + f ( x + e i ) . The operations S i commute with all discrete derivatives. Using multiindex notation(5.55) ∇ α := d Y i =1 ∇ α i i and S α := d Y i =1 S α i i , we get the Leibniz rule(5.56) ∇ γ ( f g ) = X α + β = γ C α , β (cid:0) S α ∇ β f (cid:1)(cid:0) S β ∇ α g (cid:1) , with suitable constants C α , β . Thus(5.57) B ( l ) k C k +1 ξ = L (2 l − k X | γ | = l X α + β = γ C α , β S α ( ∇ β ) ∗ ( η X ∗ ) S β ( ∇ α ) ∗ ∇ γ C k +1 ξ. Notice that k S β k = 1 (with the operator norm induced by l norms on V N ). Further,using (5.41), (4.23), and again (5.56), we have(5.58) (cid:12)(cid:12) ( ∇ β ) ∗ ( η X ∗ ) (cid:12)(cid:12) ≤ Θ C max L − k | β | with(5.59) C max = X α , β | α + β |≤ M +4 C α , β . As a result we get, recalling that l ≤ f M + 4, where f M = ⌊ d +22 ⌋ , and that η (2( f M + 4) , d ) = η ( d ) , (5.60) k B ( l ) k C k +1 k ≤≤ L (2 l − k X | γ | = l X α + β = γ C α , β Θ C max L − k | β | L ( k +1) d c max L − k ( d − | α | + l ) L η ( d ) ≤≤ Θ C c max L d + η ( d ) . This completes the proof of (ii) with M = Θ C c max .To prove the estimate (iii), we first observe that C k Λ N = 0. Hence B k C k can beviewed as an operator from V N (instead of X N ) to V N with the same trace. Tocompute the trace of B k C k +1 we now use the orthonormal basis given by the unitcoordinate vectors(5.61) e x ( z ) = (cid:26) , z = x, , z = x. According to (5.57), for l ≥ (cid:12)(cid:12) (e x , B ( l ) k C k +1 e x ) (cid:12)(cid:12) = 0 whenever x / ∈ ( X ∗ ) ∗ . For x ∈ ( X ∗ ) ∗ we use (5.57) and the bound(5.63) sup z (cid:12)(cid:12) ( ∇ α ) ∗ ∇ γ C k +1 ( z ) (cid:12)(cid:12) ≤ c max L − k ( d − | α | + | γ | ) L η ( d ) to conclude that(5.64) (cid:12)(cid:12) (e x , B ( l ) k C k +1 e x ) (cid:12)(cid:12) ≤ Θ C c max L − kd + η ( d ) and(5.65) Tr B ( l ) k C k +1 = X x ∈ Λ N (e x , B ( l ) k C k +1 e x ) ≤ Θ C c max d +2 L η ( d ) | X | k . . PROPERTIES OF THE NORMS 51 For B (0) k , we explicitly express the projection, Π e x = e x − Λ N | Λ N | , yielding(5.66) L k (e x , B (0) k C k +1 e x ) = ( Π e x , η X ∗ C k +1 e x ) == (e x , η X ∗ C k +1 e x ) − (1l Λ N | Λ N | , η X ∗ C k +1 e x )= η X ∗ ( x ) C k +1 (0) − | Λ N | (1l Λ N , η X ∗ C k +1 e x ) . Therefore(5.67) Tr B (0) k C k +1 = X x ∈ Λ N (e x , B (0) k C k +1 e x ) == L − k (cid:16) X x ∈ Λ N η X ∗ ( x ) (cid:17) C k +1 (0) − | Λ N | (1l Λ N , η X ∗ C k +1 Λ N ) ≤≤ L − k c max L − k ( d − L η ( d ) X x ∈ ( X ∗ ) ∗ ≤ c max d +2 L η ( d ) | X | k . Thus Tr (cid:16) B k C k +1 (cid:17) ≤ C (cid:12)(cid:12)(cid:0) X ∗ (cid:1) ∗ (cid:12)(cid:12) k ≤ ( M + 5) Θ C c max d +2 L η ( d ) | X | k . We get the claim (iii) with M = ( f M + 5) Θ C c max d +2 . (cid:3) Remark . Notice that, with the particular values of M and M givenabove, we can choose h fulfilling (5.49) by taking(5.68) h = C ( f M + 5) M . ⋄ HAPTER 6 Smoothness We prove Proposition 4.6 asserting the smoothness of the renormalisation map(6.1) S : U × B ⊂ ( M ( B , X ) × M ( P c , X )) × R d × d sym → M (( P ′ ) c , X )on a suitable scale of functions spaces. Here, B = B k , P = P k , and P ′ = P k +1 with k fixed. (Later, when the dependence of the map S on k will be crucial, wewill use the notation S k instead of S .) Let us recall the explicit formula (4.68) for K k +1 = K ′ = S ( H, K, q ),(6.2) K ′ ( U, ϕ ) = X X ∈P ( U ) χ ( X, U ) e I U \ X ( ϕ ) Z X (cid:0) e J ( ϕ ) ◦ P ( ϕ + ξ ) (cid:1) ( X ) µ k +1 (d ξ )with e I = e − e H , e J = 1 − e I , P = ( I − ◦ K , and I = e − H .It will be useful to split the map S into a composition of a series of mapsand to deal with them one by one. To this end, we first recall the notationfor relevant normed spaces. In Section 4.4 we have already introduced the se-quence of normed spaces M = M r ֒ → M r − ֒ → . . . ֒ → M r − m , defined as M r = { K ∈ M ( P c , X ) : k K k ( A ) k,r < ∞} and equipped with the norm k·k ( A ) k,r , r = r , r − , . . . , r − m , the space M = ( M ( B k , X ) , k·k k, ), and the se-quence of spaces M ′ = M ′ r ֒ → M ′ r − ֒ → . . . ֒ → M ′ r − m with M ′ r = { K ∈ M ( P c k +1 , X ) , k K k ( A ) r,k +1 < ∞} , equipped with the norm k·k ( A ) k +1 ,r , r = r , r − , . . . , r − m . We also introduce the space M |k = { F ∈ M ( B , X ) , |k F k| k < ∞} .One difficulty is that convolution with the measure µ k +1 does not preserve thefactorization in connected k -polymers. More precisely, if K ( X, ϕ ) = Y Y ∈C ( X ) K ( Y, ϕ )and if RK ( X, ϕ ) := Z X K ( X, ϕ + ξ ) µ k +1 (d ξ ) , then in general RK ( X, ϕ ) = Y Y ∈C ( X ) RK ( Y, ϕ )because the support of the covariance C k +1 has range bounded by L k +1 / L k / 2. Thus we cannot only consider functionals defined for connected k -polymersbut we need to consider functionals which involve all k -polymers and we define(6.3) c M r := { K ∈ M ( P k , X ) , k K k ( A , B ) k,r < ∞} , We are grateful to S. Buchholz for pointing this out and for suggesting the use of the norm k · k ( A , B ) k,r . 534 6. SMOOTHNESS (6.4) c M : ,r := { K ∈ M ( P k , X ) , k K k ( A , B ) k : k +1 ,r < ∞} , where(6.5) k K k ( A , B ) k,r := sup X ∈P k \ ∅ Γ A ( X ) B |C ( X ) | k K ( X ) k k,X,r with(6.6) Γ A ( X ) := Y Y ∈C ( X ) Γ A ( Y ) for X ∈ P \ ∅ and where k·k ( A , B ) k : k +1 ,r is defined in the same way using k K ( X ) k k : k +1 ,X,r . Note thatthe definition of the spaces does not depend on the weights A > B > S will be rewritten as a composition of several partial maps:The exponential map, E : M → M |k defined by(6.7) E ( e H ) = exp {− e H } = e I, (6.8)three polynomial maps, P : M |k × M |k × c M : ,r → M ′ r defined by(6.9) P ( e I, e J, e P )( U, ϕ ) = X X ,X ∈P ( U ) X ∩ X ∅ χ ( X ∪ X , U ) e I U \ ( X ∪ X ) ( ϕ ) e J X ( ϕ ) e P ( X , ϕ ) , (6.10) P : M |k × M r → M r defined by(6.11) P ( I, K ) = ( I − ◦ K, (6.12) P : M r → c M r , (6.13) ( P K )( X, ϕ ) = Y Y ∈C ( X ) K ( Y, ϕ )(6.14)and, finally, two linear renormalisation maps that are the source of loss of regularity, R : c M r × B → c M : ,r defined by(6.15) R ( P, q )( X, ϕ ) = ( R ( q ) P )( X, ϕ ) = Z X P ( X, ϕ + ξ ) µ ( q ) k +1 (d ξ ) , X ∈ P , (6.16) R : M × M r × B → M defined by(6.17) R ( H, K, q )( B, ϕ ) = Π (cid:16) ( R ( q ) H )( B, ϕ ) − X X ∈S X ⊃ B | X | ( R ( q ) K )( X, ϕ ) (cid:17) , (6.18)where we write B = (cid:8) q ∈ R d × d sym : k q k < (cid:9) .In terms of these maps we have(6.19) S ( H, K, q ) = P (cid:0) E ( R ( H, K, q )) , − E ( R ( H, K, q )) , R ( P ( P ( E ( H ) , K )) , q ) (cid:1) . Notice that the norms on the corresponding spaces are chosen in a natural way,with the exception of the space M ( P , X ) in the role of the domain space of themap P as well as the target space of the map R , that comes equipped with the .1. IMMERSION E : M → M |k norm k·k ( A , B ) k : k +1 ,r . This is driven by the bound (iv) from Lemma 5.1 that makes thenorm k K ( X, · ) k k : k +1 ,r natural for the map R . The additional weight B |C ( X ) | inthe norms of c M r and c M : ,r plays an important role in the estimates for the map P and is a substitute for the fact that we no longer deal with maps which factor inconnected k -polymers. More precisely if K factors we can use the bound (i) fromLemma 5.1 to conclude that k K ( X ) k k,X,r ≤ Y Y ∈C ( X ) k K ( Y ) k k,Y,r ≤ Γ A ( X ) − h k K k ( A ) k,r i |C ( X ) | . This provides additional smallness if k K k ( A ) k,r is small and the number of connectedcomponents |C ( X ) | is large. If K does not factor we can use the bound k K ( X ) k k,X,r ≤ Γ A ( X ) − B −|C ( X ) | k K k ( A , B ) k,r instead to get a good decay for a large number of components.The dependence on the parameters A and B in the definition of the weak norms(4.41) and in the norm (6.5) plays an important role here, we thus incorporate itexplicitly into the notation and write, e.g., k·k ( A ) k,r . Note that for a fixed N (where L N is the system size) the norms k·k ( A ) k,r and k·k ( A , B ) k,r are equivalent for all A > B > N . Since we are interested in bounds on thederivatives which are independent of N a careful choice of the parameters A and B is crucial.In the following sections we will show that all maps introduced above belong tothe class e C m ( X × B , Y ), introduced in Appendix D, for suitable scales of spaces X = X m ֒ → . . . ֒ → X and Y = Y m ֒ → . . . ֒ → Y . Finally we will use the chainrule in the e C m spaces to show that the same regularity for the composed map S ,see Section 6.7. In fact the maps above actually possess arbitrarily many Fr´echetderivatives (or are even real-analytic) but the setting of the e C m spaces is settingwhich naturally goes with the estimates that are independent of N (where L N isthe system size).Let us first discuss the partial maps one by one, starting from the most interiorone in the composition (6.19). E : M → M |k While the norm k H k k, is expressed directly in terms of the co-ordinates λ, a, c , d of the ideal Hamiltonian H ∈ M , the terms involving E ( H )( B, ϕ ) = e − H ( B,ϕ ) will be evaluated with the help of the norm |k·k| k . Considering thus the map E : M → M |k , we have: Lemma . We have |k H k| k ≤ k H k k, for any H ∈ M . Moreover, thereexist constants δ = δ ( r ) and C = C ( r ) so that E is smooth on B δ = { H ∈ M : k H k k, < δ } with uniformly bounded derivatives, (6.20) |k D j E ( H )( ˙ H, . . . , ˙ H ) k| k ≤ C k ˙ H k jk, , j ≤ m. In particular we have (6.21) |k E ( H ) − k| k ≤ C k H k k, . Remark . The definition of norm |k·k| k involves the parameter r (see(4.30)) but the statement does not depend on r . ⋄ Proof. Let H ∈ M and B ∈ B . First, we estimate |k H ( B, · ) k| k,B by k H k k, .In view of the definitions (4.30) and (4.25), we need to compute the norms | D p H ( B, ϕ ) | k,B , p = 0 , , , (the higher derivatives vanish as H is a quadratic function).Starting with p = 0 and recalling the definitions (4.18)–(4.20), we get(6.22) | D H ( B, ϕ ) | k,B = | H ( B, ϕ ) | ≤ | λ | L dk + L dk d X i =1 | a i | (cid:0) X x ∈ B |∇ ϕ ( x ) | (cid:1) / ++ L dk d X i,j =1 | c i,j | (cid:0) X x ∈ B |∇ ϕ ( x ) | (cid:1) / + X x ∈ B |∇ ϕ ( x ) | d X i,j =1 | d i,j | . Here, when evaluating the term P x ∈ B P di =1 | a i ||∇ i ϕ ( x ) | , we first apply the Cauchy-Schwarz inequality in R d and using the bound | a | = (cid:0)P di =1 | a i | (cid:1) / ≤ P di =1 | a i | = | a | , we then employ the Cauchy-Schwarz inequality for the second time on thesum P x ∈ B · |∇ ϕ ( x ) | with |∇ ϕ ( x ) | = P di =1 |∇ i ϕ ( x ) | . Similarly we treat thenext term with |∇ ϕ ( x ) | = P di,j =1 |∇ i ∇ j ϕ ( x ) | . In the last term we just usethe bound (cid:12)(cid:12) P di,j =1 d i,j ∇ i ϕ ( x ) ∇ j ϕ ( x ) (cid:12)(cid:12) ≤ k d k|∇ ϕ ( x ) | and then evaluate theoperator norm, k d k ≤ ( P di,j +1 d i,j ) / ≤ P di,j =1 | d i,j | .Hence,(6.23) (cid:12)(cid:12) H ( B, ϕ ) (cid:12)(cid:12) ≤≤ k H k k, (cid:16) h (cid:0) X x ∈ B |∇ ϕ ( x ) | (cid:1) / + 1 h L k (cid:0) X x ∈ B |∇ ϕ ( x ) | (cid:1) / + 1 h X x ∈ B |∇ ϕ ( x ) | (cid:17) ≤ k H k k, (cid:16) h X x ∈ B (cid:0) |∇ ϕ ( x ) | + L k |∇ ϕ ( x ) | (cid:1)(cid:17) ≤ k H k k, (cid:16) W B ( ϕ ) (cid:17) , where we took into account the definition (4.28) of the weight function W B ( ϕ ) = W Bk ( ϕ ).Similarly, taking into account that DH ( B, ϕ )( ˙ ϕ ) = ℓ ( ˙ ϕ ) + 2 Q ( ϕ, ˙ ϕ ), we get(6.24) | DH ( B, ϕ ) | k,B = sup | ˙ ϕ | k,B ≤ | ℓ ( ˙ ϕ ) + 2 Q ( ϕ, ˙ ϕ ) | ≤≤ sup {| ℓ ( ˙ ϕ ) | + | Q ( ϕ, ˙ ϕ ) | : sup x ∈ B ∗ |∇ ˙ ϕ ( x ) | ≤ hL − kd and sup x ∈ B ∗ |∇ ˙ ϕ ( x ) | ≤ hL − kd − k }≤ hL − kd (cid:8) L kd X i =1 | a i | + L kd L − k d X i,j =1 | c i,j | + d X i,j =1 | d i,j | X x ∈ B | ( ∇ ϕ )( x ) (cid:1) | (cid:9) ≤≤ k H k k, (cid:0) h (cid:0)X x ∈ B |∇ ϕ ( x ) | (cid:1) / (cid:1) ≤ k H k k, (cid:0) W B ( ϕ ) (cid:1) .1. IMMERSION E : M → M |k and(6.25) | D H ( B, ϕ ) | k,B ≤ h L − dk L dk d X i,j =1 | d i,j | ≤ k H k k, . Recalling that D H ( B, ϕ )( ˙ ψ, ˙ ψ, ˙ ψ ) = 0, we finally get(6.26) |k H k| k = |k H ( B, · ) k| k,B ≤ ϕ W − Bk ( ϕ ) k H k k, (1 + log W B ( ϕ )) ≤ k H k k, . To get |k E ( H ) k| k , we need to compute the norms | D p E ( H )( B, ϕ ) | k,B , p =0 , . . . , r . Using again Fa`a di Bruno’s chain rule for higher order derivatives andthe bounds (6.23), (6.24), and (6.25), we get(6.27) | D p E ( H )( B, ϕ ) | k,B ≤ B r e − H ( B,ϕ ) (cid:16) k H k k, (1 + log W B ( ϕ )) (cid:17) p with the constant B r ≤ r r bounding the number of partitions of the set { , . . . , p } .Hence,(6.28) |k E ( H ) k| k ≤ B r sup ϕ e − H ( B,ϕ ) W − B ( ϕ ) r X p =0 (cid:0) k H k k, (1 + log W B ( ϕ )) (cid:1) p ≤≤ B r r X p =0 sup ϕ e k H k k, (1+log W B ( ϕ )) W − B ( ϕ )e p k H k k, (1+log W B ( ϕ )) ≤≤ ( r + 1) B r e r ) k H k k, sup ϕ e k H k k, (1+ r ) log W B ( ϕ ) W − B ( ϕ ) < e ( r + 1) B r once k H k k, is sufficiently small to assure that 2 k H k k, (1 + r ) ≤ W B ( ϕ ) ≥ E ( H ) as a composed function, we get DE ( H )( ˙ H )( B, ϕ ) = E ( H )( B, ϕ ) ˙ H ( B, ϕ ). Using, similarly as when proving (5.1),the fact that a Taylor expansion of a product is the product of Taylor expansions,we get(6.29) | DE ( H )( ˙ H )( B, ϕ ) | k,B,r ≤ | E ( H )( B, ϕ ) | k,B,r | ˙ H ( B, ϕ ) | k,B,r . Applying now (6.27) and (6.23)–(6.25), we get(6.30) | DE ( H )( ˙ H )( B, ϕ ) | k,B,r ≤≤ e − H ( B,ϕ ) ( r + 1) (cid:0) k H k k, (1 + log W B ( ϕ )) (cid:1) r k ˙ H k k, (1 + log W B ( ϕ ))yielding |k DE ( H )( ˙ H ) k| k (6.31) ≤ sup ϕ e − H ( B,ϕ ) ( r + 1) W − B ( ϕ )e r k H k k, (1+log W B ( ϕ )) k ˙ H k k, e log W B ( ϕ ) ≤ 10 e( r + 1) k ˙ H k k, if 4 r k H k k, ≤ 1. Similarly, we get the bounds for higher derivatives. Formally theestimate (6.21) follows from (6.20) and the identity E ( H ) − Z DE ( tH )( H ) d t. (cid:3) P Lemma . Consider the map P : M |k × M r → M r defined in (6.12) , re-stricted to B ρ (1) × B ρ ⊂ M |k × M r with the balls B ρ (1) = { I : |k I − k| k < ρ } and B ρ = { K : k K k ( A ) k,r < ρ } and the target space M r equipped with the norm k·k ( A / k,r . For any A ≥ and ρ , ρ such that (6.32) ρ < (2 A ) − , and ρ < (2 A d ) − , the map P restricted to B ρ × B ρ is smooth and satisfies the bound (6.33) 1 j ! j ! (cid:13)(cid:13)(cid:0) D j D j P )( I, K )( ˙ I, . . . , ˙ I, ˙ K, . . . , ˙ K ) (cid:13)(cid:13) ( A / k,r ≤ (2 A ) j (cid:0) A d (cid:1) j |k ˙ I k| j k (cid:0) k ˙ K k ( A ) k,r (cid:1) j for any j , j ∈ N . In particular, (6.34) k P ( I, K ) k ( A / k,r ≤ A |k I − k| k + 2 A d k K k ( A ) k,r . Proof. Recall that(6.35) (cid:0) ( I − ◦ K (cid:1) ( X ) = X Y ∈P ( X ) ( I − X \ Y K ( Y ) , X ∈ P c , with ( I − X \ Y = Q B ∈B ( X \ Y ) (cid:0) I ( B ) − (cid:1) and K ( Y ) = Q Z ∈C ( Y ) K ( Z ), where C ( Y )denotes the set of components of Y ∈ P .Hence,(6.36) 1 j ! j ! (cid:0) D j D j (cid:0) ( I − ◦ K (cid:1) ( X )( ˙ I, . . . , ˙ I, ˙ K, . . . , ˙ K ) == X Y ∈P ( X ) ,Y ∈P ( X \ Y ) , | Y | = j J ⊂C ( Y ) , |J | = j ( I − ( X \ Y ) \ Y ˙ I Y Y Z ∈C ( Y ) \J K ( Z ) Y Z ∈J ˙ K ( Z ) . Further, recall that, by definition of the norm k K k ( A ) k,r , we have k K ( Z ) k k,Z,r ≤ Γ A ( Z ) − k K k ( A ) k,r for any Z ∈ P c k . Notice also that(6.37) A | Z |− d ≤ max(1 , A | Z |− d ) ≤ Γ A ( Z ) ≤ A | Z | for any A ≥ Z ∈ P c . Using the bounds (iia) and (i) from Lemma 5.1,assumptions (6.32), as well as the lower bound on Γ A ( Z ) above and the fact thatthe number of terms in the sum is bounded by 2 | X | , we get(6.38) k P ( I, K )( X ) k k,X,r ≤≤ X Y ∈P ( X ) |k I − k| | X \ Y | k (cid:0) k K k ( A ) k,r (cid:1) |C ( Y ) | A d |C ( Y ) | A −| Y | ≤ A −| X | | X | = (cid:0) A (cid:1) −| X | , .3. THE MAP P cf. [ Bry09 , Lemma 6.3]. Similarly, using that (cid:0) nj (cid:1) ≤ n , we get the claim(6.39)1 j ! j ! k ( D j D j P )( I, K )( X )( ˙ I, . . . , ˙ I, ˙ K, . . . , ˙ K ) k k,X,r ≤ X Y ∈P ( X ) (cid:0) | X \ Y | j (cid:1) |k I − k| | X \ Y |− j k |k ˙ I k| j k (cid:0) |C ( Y ) | j (cid:1) × (cid:0) k K k ( A ) k,r (cid:1) |C ( Y ) |− j (cid:0) k ˙ K k ( A ) k,r (cid:1) j A d C ( Y ) A −| Y | ≤ X Y ∈P ( X ) | X \ Y | (2 A ) − ( | X \ Y |− j ) |k ˙ I k| j k |C ( Y ) | (2 A d ) − ( |C ( Y ) |− j ) (cid:0) k ˙ K k ( A ) k,r (cid:1) j × A d C ( Y ) A −| Y | == X Y ∈P ( X ) j A − ( | X \ Y |− j ) |k ˙ I k| j k j A j d (cid:0) k ˙ K k ( A ) k,r (cid:1) j A −| Y | ≤ | X | (2 A ) j |k ˙ I k| j k (2 A d ) j (cid:0) k ˙ K k ( A ) k,r (cid:1) j A −| X | . Finally, (6.34) follows from the fact that P (1 , 0) = 0 and(6.40) dd t P (1 + t ( I − , tK ) = D P (1 + t ( I − , tK )( I − D P (1 + t ( I − , tK ) K. (cid:3) P Lemma . Let A ≥ , B ≥ . Consider the map P : M r → c M r defined by (6.41) ( P K )( X ) = Y Y ∈C ( X ) K ( Y ) . restricted to B ρ = { K ∈ M r : k K k ( A ) k,r < ρ } and the target space c M r equipped withthe norm k·k ( A , B ) k,r . For any (6.42) ρ ≤ (2 B ) − the map P restricted to B ρ is smooth and satisfies the bound (6.43) 1 j ! (cid:13)(cid:13)(cid:0) D j P )( K )( ˙ K, . . . , ˙ K ) (cid:13)(cid:13) ( A , B ) k,r ≤ (cid:0) B k ˙ K k ( A ) k,r (cid:1) j for any j , j ∈ N . Proof. The proof is similar to, but simpler than, the proof of Lemma 6.3. Wehave(6.44) 1 j ! D j P ( K )( X )( ˙ K, . . . , ˙ K ) = X J ⊂C ( X ) , |J | = j Y Z ∈C ( X ) \J K ( Z ) Y Z ∈J ˙ K ( Z ) . Thus using the estimate (cid:0) |C ( X ) j (cid:1) ≤ |C ( X ) and the identity Γ A ( X ) = Q Z ∈C ( X ) Γ A ( Z )and arguing as in the proof of Lemma 6.3 we get(6.45) B |C ( X ) | Γ A ( X ) 1 j ! k D j P ( K )( X )( ˙ K, . . . , ˙ K ) k k,X,r ≤ (2 B ) |C ( X ) | (cid:16) k K k ( A ) k,r (cid:17) |C ( X ) |− j ×× (cid:16) k ˙ K k ( A ) k,r (cid:17) j . Since 2 B k K k ( A ) k,r ≤ B ρ ≤ j ! k D j P ( K )( X )( ˙ K, . . . , ˙ K ) k ( A , B ) k,r ≤ (cid:16) B k ˙ K k ( A ) k,r (cid:17) j and this finishes the proof. (cid:3) R Lemma . Let m ∈ N , m ≤ r , and for any n = 0 , , . . . , m , let X n denotethe space c M r − m +2 n equipped with the norm k·k X n = k·k ( A , B ) k,r − m +2 n and Y n thespace c M : ,r − m +2 n equipped with the norm k·k Y n = k·k ( A / , B / d ) k : k +1 ,r − m +2 n . Further, let B = { q ∈ R d × d sym : k q k < } . Consider the map R : X × B → Y defined in (6.16) with X = X m = c M r and Y = Y m = c M : ,r . There exists a constant C = C ( r , d ) such that for any h ≥ L κ ( d ) h with h = h ( d, ω ) and κ ( d ) as in Lemma 5.1 (iv)(see (5.68) ), A ≥ , and any r = 1 , . . . , r , we have (6.47) R ∈ e C m ( X × B , Y ) . Moreover the constants in the estimates of the relevant derivatives are independentof k and N . More precisely for ≤ ℓ ≤ m, ≤ n ≤ m − ℓ , there are C ( n, d ) > such that k D ℓ R ( P, q , ˙ q ℓ ) k Y n ≤ C ( n, d ) k P k X n + ℓ k ˙ q k ℓ , (6.48) k D D ℓ R ( P, q , ˙ P , ˙ q ℓ ) k Y n ≤ C ( n, d ) k ˙ P k X n + ℓ k ˙ q k ℓ , (6.49) D D ℓ R ( P, q , ˙ P , ˙ q ℓ ) = 0 . (6.50) Remark . (i) Note that (6.49) follows from (6.48) since R is linear in thefirst argument, whereas (6.50) is trivial.(ii) The proof below actually shows that(6.51) k D ℓ R ( P, q , ˙ q ℓ )( X ) k k : k +1 ,X,n ≤ C ( n, d )2 | X | k P ( X ) k k,X,n + ℓ k ˙ q k ℓ The estimate (6.48) then follows by the choice of weights A / B / d on thetarget space, see Step 2 of the proof.(iv) It follows from Step 1 in the proof, the bound k R ( q ) P ( X ) k k : k +1 ,X,r ≤ | X | k k P ( X ) k k,X,r in Step 2 of the proof and the linearity of R in the first argument that R isactually a real-analytic map from X r × B to Y r without any loss of regularity.The bounds on the corresponding derivatives depend, however, on the system size N and the level k , while the bounds stated in Lemma 6.5 do not. ⋄ .4. THE MAP R Proof. Recall from (6.16) that R ( P, q )( X, ϕ ) = ( R ( q ) P )( X, ϕ ) = Z X P ( X, ϕ + ξ ) µ ( q ) k +1 (d ξ ) . The fact that R maps M m × B to Y m follows from Lemma 5.1(iv). Note that R is linear in P . Thus by Lemma D.31 it suffices to show that(i) For each P ∈ X m and 0 ≤ ℓ ≤ m the map q R ( q ) P is in C ℓ ∗ ( B ; Y m − ℓ ).(ii) For each q ∈ B there exist δ, C > k D ℓ q R ( q ) ( P, q ) , ˙ q ℓ ) k Y n ≤ C k P k X n + ℓ k ˙ q k ℓ for any 0 ≤ ℓ ≤ m, ≤ n ≤ m − ℓ , and for all ( P, q , ˙ q ) ∈ X m × B δ ( q ) × R d × d sym .We split the proof of (i) and (ii) into seven steps below. Note that the requiredconstant C will be given as the maximum of all constants in (6.48) and (6.49). Wefirst show (i) in step 1 below. Indeed we even show that q R ( q ) P is real-analyticwith values in Y m ⊂ Y m − ℓ . Step 1: Assume that P ∈ M r = ( M r ( P c k , X ) , k·k ( A ) k,r ) for some r ∈ { r , . . . , r − m } . Then the map q R ( P, q )is real-analytic from B to M : ,r = ( M r ( P c k , X ) , k·k ( A / k : k +1 ,r ). First it suffices to showthe result for r = 0, since differentiation with respect to ϕ commutes with R ( q ) .Secondly it suffices to consider a fixed polymer X , since there are only finitely manypolymers. Thus we need to show the following: If k P ( X ) k k,X, = sup ξ | P ( X, ξ ) | w Xk ( ξ ) < ∞ , then the map B ∋ q Z X P ( X, · + ξ ) µ C ( q ) k +1 (d ξ )is real-analytic with values in the space of continuous functions F of the field withthe weighted norm k F k k : k +1 ,X, = sup ϕ | F ( ϕ ) | w Xk : k +1 ( ϕ ) . This follows from Gaussian calculus (see Lemma C.1), Lemma 5.3 and the propertiesof the finite range decomposition, see Proposition 4.1. To see this recall (5.42), i.e., w Xk ( ϕ + ξ ) ≤ w Xk : k +1 ( ϕ )e κ ( B k ξ,ξ ) , where κ = 2 Ch − and B k is given by (5.43). If h and κ ( d ) are chosen as inLemma 5.1 and h ≥ L κ ( d ) h then it follows from Lemma 5.3 that for q ∈ B and C k +1 = C ( q ) k +1 we have(6.52) 0 ≤ C / k +1 κ B k C / k +1 ≤ 12 Id and hence C − k +1 > κ B k , i.e., B ∋ q 7→ U k , where we define U k := { C ∈ Sym (+) ( X ) : C − > κ B k } . By Lemma C.1 the map C Z X P ( · + ξ ) µ C (d ξ )is real-analytic from U k to the desired space. Finally, by Proposition 4.1 and (6.52)the map q C ( q ) k +1 is real-analytic from B to U k .Hence q R ( P, q ) is real-analytic from B to the space M : ,r , and thus (i)is proven.In the remaining steps we are going to prove (ii). In step 2 we show the boundsfor ℓ = 0 followed by the bound for ℓ = 1 in step 3 to step 6. The bounds for higherderivatives are then finally settled in step 7. Step 2: Bounds on R ( q ) . By Lemma 5.1(iv) we have for all q ∈ B the followingestimate k R ( q ) P ( X ) k k : k +1 ,X,r ≤ | X | k k P ( X ) k k,X,r . For connected polymers Y we have(6.53) 2 | Y | Γ A / ( Y ) ≤ d Γ A ( Y ) . Thus for general polymer X we get(6.54) 2 | X | Γ A / ( X ) ≤ d |C ( X ) | Γ A ( X ) . and thus(6.55) 2 | X | (cid:16) B / d (cid:17) |C ( X ) | Γ A / ( X ) ≤ B |C ( X ) | Γ A ( X ) . Therefore k R ( q ) P k ( A / , B / d ) k : k +1 ,r ≤ k P k ( A , B ) k,r , and hence with r = r − m + 2 n we obtain(6.56) k R ( P, q ) k Y n = k R ( q ) P k Y n ≤ k P k X n , for all q ∈ B . Step 3: Bounds for D R ( P, q, ˙ q ). Let q ∈ B and k ˙ q k = 1 and write γ ( t ) = q + t ˙ q in the following. By Lemma C.2 and (C.23) we have D R ( P, q , ˙ q )( X, ϕ ) = dd t (cid:12)(cid:12)(cid:12) t =0 Z X P ( X, ϕ + ξ ) µ C ( γ ( t )) k +1 (d ξ )= Z X A ˙ C k +1 P ( X, ϕ + ξ ) µ C ( q ) k +1 (d ξ ) = ( R ( q ) A ˙ C k +1 P )( X, ϕ )with ˙ C k +1 = dd t (cid:12)(cid:12)(cid:12) t =0 C ( γ ( t )) k +1 and where the functional A ˙ C k +1 is defined as A ˙ C k +1 P ( X, ξ ) = L dN − X i,j =1 D P ( X, ξ, e i , e j )( ˙ C k +1 ) i,j , where { e j } L dN − j =1 is any orthonormal basis of X and ( ˙ C k +1 ) i,j = ( ˙ C k +1 e i , e j ). ByStep 2 we obtain the following bound for the derivative with respect to q , for0 ≤ n ≤ m − k D R ( P, q , ˙ q ) k Y n ≤ k A ˙ C k +1 P k X n . .4. THE MAP R Step 4: Estimate for k A ˙ C k +1 P k . We now express and estimate the functional A ˙ C k +1 P using the orthonormal Fourier basis { f p } p ∈ b T N of the (complexified space) X given by(6.58) f p ( x ) = e i h p,x i L dN/ , p ∈ b T N , x ∈ Λ N . We denote by ˙ [ C k +1 ( p ) the Fourier multiplier of ˙ C k +1 . Now C ( q ) k +1 and hence ˙ C k +1 are diagonal in the Fourier basis and˙ C k +1 f p = ˙ [ C k +1 ( p ) f p with ˙ [ C k +1 ( p ) ∈ R . Thus by (C.13) A ˙ C k +1 P ( X, ξ ) = X p ∈ b T N D P ( X, ξ, ˙ C k +1 f p , f p ) X p ∈ b T N D P ( X, ξ, f p , f p ) ˙ [ C k +1 ( p ) . We claim that(6.59) | A ˙ C k +1 D P ( X, ξ ) ˙ C k +1 ) | k,X,r − ≤ r ( r − | P ( X, ξ ) | k,X,r X p ∈ b T N \{ } | f p | k,X | ˙ [ C k +1 | ( p ) . whenever ˙ C k +1 is diagonal in the Fourier basis. In particular we now show thatthere exists a C ( n, d ) > ≤ n ≤ m − k A ˙ C k +1 P k X n ≤ C ( n ) k P k X n +1 X p ∈ b T N | f p | ˙ [ C k +1 ( p ) . Indeed, using the fact that ˙ [ C k +1 ( p ) is real and the definition of the trace wehave(6.61) G ( X, ξ ) :=Tr (cid:0) D P ( X, ξ ) ˙ C k +1 ) (cid:1) = A ˙ C k +1 P ( X, ξ )= X p ∈ b T N \{ } (cid:0) f p , D P ( X, ξ ) f p (cid:1) ˙ [ C k +1 ( p )= X p ∈ b T N \{ } (cid:0) Re ( f p ) , D P ( X, ξ )Re ( f p ) (cid:1) ˙ [ C k +1 ( p )+ X p ∈ b T N \{ } (cid:0) Im ( f p ) , D P ( X, ξ )Im ( f p ) (cid:1) ˙ [ C k +1 ( p ) . By a standard symmetrisation argument we have(6.62) | D σ P ( X, ξ )( ˙ ϕ , . . . , ˙ ϕ σ ) | ≤ σ σ σ ! | D σ P ( X, ξ ) | k,X σ Y i =1 | ˙ ϕ i | k,X . Set(6.63) M := X p ∈ b T N \{ } | f p | k,X | ˙ [ C k +1 | ( p ) . Then for all ˙ ϕ with | ˙ ϕ | k,X ≤ | D s G ( X, ξ )( ˙ ϕ, . . . , ˙ ϕ ) | ≤≤ X p ∈ b T N \{ } | D s +2 P ( X, ξ )( ˙ ϕ, . . . , ˙ ϕ ; Re ( f p ) , Re ( f p )) || ˙ [ C k +1 | ( p )+ X p ∈ b T N \{ } | D s +2 P ( X, ξ )( ˙ ϕ, . . . , ˙ ϕ ; Im ( f p ) , Im ( f p )) || ˙ [ C k +1 | ( p ) ≤ s + 2) s +2 ( s + 2)! | D s +2 P ( X, ξ ) | k,X,r | ˙ ϕ | sk,X M. Hence | D s G ( X, ξ ) | k,X ≤ C ( r ) M | D s +2 P ( X, ξ, X ) | k,X , for all s ≤ r − C ( r ) = 2 r r r ! . This yields(6.65) | G ( X, ξ ) | k,X,r − ≤≤ M r − X s =0 s ! | D s +2 P ( X, ξ ) | k,X ≤ r ( r − C ( r ) M r − X s =0 s + 2)! | D s +2 P ( X, ξ ) | k,X ≤≤ r ( r − C ( r ) M | P ( X, ξ ) | k,X,r and hence the assertion (6.60). Note that in the proof we only used the fact that C ( γ ( t )) k +1 is diagonal in the Fourier basis. Hence the same computation yields thecorresponding result for the higher derivatives(6.66) | Tr( D P ( X, ξ ) d j d t j C γ ( t ) k +1 ) | k,X,r − ≤≤ r ( r − C ( r ) | P ( X, ξ ) | k,X,r X p ∈ b T N \{ } | f p | k,X (cid:12)(cid:12)(cid:12)(cid:12) d j d t j \ C ( γ ( t )) k +1 ( p ) (cid:12)(cid:12)(cid:12)(cid:12) . Step 5: Estimate for the term (6.63) involving the Fourier multiplier. Let γ ( t ) = q + t ˙ q with q ∈ B and k ˙ q k = 1 . We claim that, with our choice of h , there exists C = C ( n, d ) > X p ∈ b T N \{ } | f p | k,X (cid:12)(cid:12)(cid:12)(cid:12) d j d t j \ C ( γ ( t )) k +1 ( p ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cj ! . To see this note first that by the definition of the | · | k,X norm(6.68) | f p | k,X ≤ h L Nd/ L kd/ max( | p | , L k | p | , L k | p | ) . The estimate (4.7) in Remark 4.2 can be rewritten as(6.69) X p ∈ b T N \{ } | p | n (cid:12)(cid:12)(cid:12)(cid:12) d j d t j \ C ( γ ( t )) k +1 ( p ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C j j ! L η ( n,d )+ n + d − L − k ( n + d − L dN , where η ( n, d ) = max( ( d + n − , d + n + 6) + 10. Applying this estimate with n = 2 , η ( n, d ) in n , we need a bound on η (6 , d ) + 4 + d . It turns out that η (6 , d ) + 4 + d ≤ κ ( d ) whenever d ≥ 2. Indeed, .4. THE MAP R this amounts to showing that η (6 , d ) + 4 ≤ η (12 , d ) (with 2 ⌊ d +22 ⌋ + 8 = 12 for d = 2). Using this and assuming that h ≥ 1, we can conclude that(6.70) h − L η ( n,d )+ n + d − ≤ n = 2 , , 6, implying thus (6.67). Step 6: Estimate for D R ( P, q, ˙ q ). It follows from Step 3, (6.60) with˙ C k +1 = dd t (cid:12)(cid:12) t =0 C ( γ ( t )) k +1 , and Step 5 with j = 1 for any 0 ≤ n ≤ m − C ( n, d ) > k D R ( P, q , ˙ q ) k Y n ≤ k A ˙ C k +1 P k X n ≤ C ( n, d ) k P k X n +1 . Step 7: Bounds for the higher derivatives D ℓ R ( P, q, ˙ q ℓ ). These boundsfollow from Gaussian calculus in Lemma C.4, the chain rule and the estimates for d j d t j C ( γ ( t )) k +1 (see step 5). We consider first the case ℓ = 2. As in (C.1) in appendix Cwe set H ( C )( · ) = Z X P ( X, · + ξ ) µ C (d ξ ) , respectively, e h ( t )( · ) = Z X P ( X, · + ξ ) µ C γ ( t ) k +1 (d ξ ) . By Lemma C.4 and (C.24) we obtain D R ( P, q , ˙ q , ˙ q )( X, ϕ ) = d d t (cid:12)(cid:12)(cid:12) t =0 R ( P, γ ( t ))( X, ϕ ) = D H ( C k +1 , ˙ C k +1 , ˙ C k +1 )+ DH ( C , ¨ C k +1 ) = R ( A C k +1 P, q )( X, ϕ ) + R ( A ¨ C k +1 P, q )( X, ϕ )where we use that˙ C k +1 = dd t (cid:12)(cid:12)(cid:12) t =0 C ( γ ( t )) k +1 and ¨ C k +1 = d d t (cid:12)(cid:12)(cid:12) t =0 C ( γ ( t )) k +1 . By step 2 we have the estimate k D R ( P, q , ˙ q , ˙ q ) k X n ≤ (cid:0) k A C k +1 P k X n + k A ¨ C k +1 P k X n (cid:1) . Now step 4 and step 5 yield the following bound, for 0 ≤ n ≤ m − k A ¨ C k +1 P k X n ≤ C ( n ) k P k X n +1 ≤ C ( n ) k P k X n +2 . Applying now the steps 4 and 5 twice we get that k A C k +1 P k X n ≤ C ( n ) k A ˙ C k +1 P k X n +1 ≤ C ( n ) k P k X n +2 , and thus the required estimate for the second derivative D R . For general ℓ ≥ D ℓ R ( P, q , ˙ q ℓ )is a linear combination of terms of the form R ( A ˙ C · · · A ˙ C κ P, q )where ˙ C i := d j i d t j i (cid:12)(cid:12)(cid:12) t =0 C ( γ ( t )) k +1 with κ X i =1 j i = ℓ. Thus the desired estimate follows from step 2 and a κ -fold application of (6.60) andstep 5. (cid:3) R Lemma . Let m ∈ N , m + 2 ≤ r . For n = 0 , , . . . , m let Z n denotethe space M r − m +2 n equipped with the norm k·k Z n = k·k ( A ) k,r − m +2 n . Let X n = M × Z n , Y n = M (for all n ) and B = { q ∈ R d × d sym : k q k < } . Consider themap R : M × X × B → Y , defined in (6.18) with X = X m = M × M r and Y = Y m = M . There exists a constant C = C ( d ) such that for any h ≥ L κ ( d ) h with h = h ( d, ω ) and κ ( d ) as in Lemma 5.1 (iv), A ≥ , we have (6.72) R ∈ e C m ( X × B , Y ) . Moreover for any q and ˙ q with | q | < and | ˙ q | ≤ , and any ℓ ≤ m , we have (6.73) k D j D n D ℓ R ( H, K, q )( ˙ H, ˙ K, ˙ q , . . . , ˙ q ) k k, ≤ C k H k k, + k K k Z ℓ if j = 0 , n = 0 , k ˙ H k k, if j = 1 , n = 0 , k ˙ K k Z ℓ if j = 0 , n = 1; and (6.74) D j D m D ℓ R ( H, K, q ) = 0 if j + m ≥ . Remark . It follows from Remark 6.6 and Lemma 6.9 below that the map R is actually a real analytic map from M × M to M . ⋄ First, we estimate the main component of R , namely the map Π . Lemma . Let B ∈ B k , X ∈ S k with X ⊃ B , and let K ∈ M ( P k , X ) . Then (6.75) k Π K ( X, · ) k k, ≤ [2 d ( d + d ) + d ] | K ( X, | k,X, . Note that since X ∈ S k we have X ⊂ B ∗ and thus the maps ϕ K ( X, ϕ ) canbe viewed as an element of M ∗ ( B , X ) on which the projection Π was defined. Proof. Let H = Π K ( X, · ). By definition we have H ( B, ˙ ϕ ) = L dk λ + ℓ ( ˙ ϕ ) + Q ( ˙ ϕ, ˙ ϕ ), where ℓ ( ˙ ϕ ) = X x ∈ B d X i =1 a i ∇ i ˙ ϕ + c i,j ∇ i ∇ j ˙ ϕ ( x )(6.76) Q ( ˙ ϕ, ˙ ϕ ) = 12 X x ∈ B d X i,j =1 d i,j ∇ i ˙ ϕ ( x ) ∇ j ˙ ϕ ( x )(6.77)and L dk λ = K ( X, ℓ ( ˙ ϕ ) = DK ( X, ϕ ) ∀ ˙ ϕ quadratic + affine in ( B ∗ ) ∗ (6.79) Q ( ˙ ϕ, ˙ ϕ ) = 12 D K ( X, ϕ, ˙ ϕ ) ∀ ˙ ϕ affine in ( B ∗ ) ∗ (6.80) .5. THE MAP R To estimate d i,j and a i we consider functions ˙ ϕ which are linear on (( B ∗ ) ∗ ) ∗ (6.81) ˙ ϕ = d X i =1 η i π i , where η = ( η i ) i =1 ,...,d ∈ R d , and π i is the co-ordinate projection π i ( x ) = x i for x ∈ Z d . Then for x ∈ ( B ∗ ) ∗ we have ∇ i ˙ ϕ ( x ) = η i and ∂ α ˙ ϕ ( x ) = 0 if | α | = 2 or | α | = 3. Hence,(6.82) L dk | d X i,j =1 d i,j η i η j | = | Q ( ˙ ϕ ) | = (cid:12)(cid:12)(cid:12) D K ( X, ϕ, ˙ ϕ ) (cid:12)(cid:12)(cid:12) ≤ | D K ( X, | k,X | ˙ ϕ | k,X = 12 | D K ( X, | k,X h − d X i =1 | η i | L dk . This yields max | η | =1 | P di,j =1 d i,j η i η j | ≤ h − | D K ( X, | k,X and thus(6.83) d X i,j =1 | d i,j | ≤ d d X i,j =1 | d i,j | ≤ d (cid:0) λ max ( d ) (cid:1) / ≤ d h − | D K ( X, | k,X . Similarly, we have(6.84) L dk d X i =1 a i η i = ℓ ( ˙ ϕ ) = DK ( X, ϕ ) ≤ | DK ( X, | k,X h − d X i =1 | η i | ! L dk . The choice η i = a i yields(6.85) d X i =1 | a i | ≤ d d X i =1 | a i | ! ≤ d h − L − dk | DK ( X, | k,X . For the evaluation of the second derivative we use a test function which satisfies(6.86) ˙ ϕ ( x ) = 12 d X i,j =1 η i,j ( x − x ) i ( x − x ) j ∀ x ∈ (( B ∗ ) ∗ ) ∗ , where x = | B | P x ∈ B x and η i,j = η j,i . Then, for any x ∈ ( B ∗ ) ∗ ,(6.87) ∇ j ˙ ϕ ( x ) = d X i =1 η i,j ( x − x ) i , ∇ i ∇ j ˙ ϕ ( x ) = η i,j , and ∇ α ˙ ϕ ( x ) = 0 for | α | = 3 . Now | ( x − x ) i | ≤ d +1 − L k ≤ d L k for any x ∈ ( B ∗ ) ∗ and thus |∇ j ˙ ϕ ( x ) | ≤ d ( P di =1 | η i,j | ) d L k which yields(6.88) | ˙ ϕ | k,B ≤ h (cid:0) d d L k L kd ( d X i,j =1 | η i,j | ) + | L k ( d +1) (cid:1) ( d X i,j =1 | η i,j | ) ≤≤ (2 d d + 1) h − L k ( d +1) ( d X i,j =1 | η i,j | ) . Note that P x ∈ B η i,j ( x − x ) i a i vanishes in view of the definition of x . Hence(6.89) d X i,j =1 L dk η i,j c i,j == ℓ ( ˙ ϕ ) ≤ | DK ( X, | k,X | ˙ ϕ | k,X ≤ (2 d d +1) h − L k ( d +1) ( d X i,j =1 | η i,j | ) | DK ( X, | k,X Taking η i,j = c i,j we get(6.90) d X i,j =1 | c i,j | ≤ d d X i,j =1 | c i,j | ≤ (2 d d + d ) h − L − ( d − k | DK ( X, | k,X . This yields the assertion with(6.91) C ( d ) = max(1 , d + 2 d ( d + d ) , d ) = d + 2 d ( d + d ) . (cid:3) Proof of Lemma 6.7. We first note that R ( H, K, q ) = R ( q )2 ,a H + R ( q )2 ,b K where R ( q )2 ,a and R ( q )2 ,b are linear maps. Thus (6.74) is obvious. To prove the remainingstatements we can consider the maps H R ( q )2 ,a H and K R ( q )2 ,b K separately. Wewill establish the relevant estimates for the directional derivatives t R ( q + t ˙ q ) ,a and t R ( q + t ˙ q ) ,b . The assertion on the existence and continuity of the total derivativesthen follows as in the proof of Lemma 6.5, using in particular the continuity of themap q R ( q ) . We first consider the map(6.92) R ( q )2 ,a H := Π R ( q ) H which acts on ideal Hamiltonians. The integral of an odd functions against µ ( q ) k +1 iszero and(6.93) Z X Q ( ξ, ξ ) µ ( q ) k +1 = L dk X i,j d i,j ∇ i ∇ ∗ j C ( q ) k +1 (0)(cf. (4.81)). Thus R ( q ) H is again an ideal Hamiltonian and the action of R ( a )2 inthe coordinates ( λ, a, c , d ) for H is simply(6.94) ( λ, a, c , d ) ( λ + X i,j d i,j ∇ i ∇ ∗ j C ( q ) k +1 (0) , a, c , d )By (4.3) we have |∇ i ∇ ∗ j C ( q ) k +1 (0) | ≤ C ( d ) L η (2 ,d ) L − dk and thus(6.95) k R ( q )2 ,a H k ≤ (1 + C ( d ) h − L η (2 ,d ) ) k H k k, ≤ C ( d ) k H k k, , .5. THE MAP R where we used the lower bound on h in the assumption of the lemma. The estimatesfor D ℓq R ( q )2 ,a H follow in the same way from (4.3) since h ≥ L κ ( d ) ≥ L η (8 ,d ) .Now let X ∈ S : k with X ⊃ B and let K ∈ M ( P k , X ). We will estimate(6.96) Π R ( q ) K ( X, · )and its derivatives with respect to q . The operator R ( q )2 ,b is obtained by taking asum over all such X (for a fixed block B ) with weight | X | k . Since there are at most(3 d − d such polymers X is suffices to estimate (6.96).By Lemma 6.9 and Lemma 5.1 (iv) we have(6.97) 1 C ( d ) k Π R ( q ) K ( X, · ) k k, ≤ | R ( q ) K ( X, | k,X, ≤ Z X | K ( X, ξ ) | k,X, µ ( q ) k +1 ( dξ ) ≤ | X | k k K ( X ) k k,X, ≤ d k K ( X ) k k,X, ≤ d k K k ( A ) k, . The derivatives with respect to q are estimated using Gaussian calculus andthe estimates used in the proof of Lemma 6.5. Let k q k < and k ˙ q k = 1, andconsider the curve γ ( t ) = q + t ˙ q on a sufficiently small interval ( − a, a ). Let(6.98) G ( X, ϕ ) := Tr (cid:2) D K ( X, ϕ ) ˙ C ( q ) k +1 (cid:3) . Then (see Appendic C)(6.99) dd t (cid:12)(cid:12)(cid:12) t =0 ( R ( γ ( t )) K )( X, ϕ ) = ( R ( q ) G )( X, ϕ )Now by (6.66) and (6.67) as well as the assumption on h we have | G ( X, ϕ ) | k,X, ≤ C | K ( X, ϕ ) | k,X, . Using again Lemma 6.9 and Lemma 5.1 (iv) we get(6.100) 1 C ( d ) k D q Π R ( q ) K ( X, · )( ˙ q ) k k, = 1 C ( d ) (cid:13)(cid:13)(cid:13)(cid:13) dd t (cid:12)(cid:12)(cid:12) t =0 Π R ( γ ( t )) K ( X, · ) (cid:13)(cid:13)(cid:13)(cid:13) k, ≤ | ( R ( q ) G )( X, | k,X, ≤ d k G ( X ) k k,X, ≤ C d k K ( X ) k k,X, ≤ C d k K k ( A ) k, . The higher derivatives with respect to t are estimated in a similar way usingthe functions G ( X, ϕ ) := Tr (cid:2) D K ( X, ϕ )¨ C ( q ) k +1 (cid:3) , G ( X, ϕ ) := Tr (cid:2) D G ( X, ϕ ) ˙ C ( q ) k +1 (cid:3) , (6.101) G ( X, ϕ ) := Tr (cid:2) D K ( X, ϕ )... C ( q ) k +1 (cid:3) , G ( X, ϕ ) := Tr (cid:2) D G ( X, ϕ )¨ C ( q ) k +1 (cid:3) , (6.102) G ( X, ϕ ) := Tr (cid:2) D G ( X, ϕ ) ˙ C ( q ) k +1 (cid:3) . (6.103)and the estimates (see (6.66) and (6.67)) | G ( X, ξ ) | k,X, + | G ( X, ξ ) | k,X, ≤ C | K ( X, ξ ) | k,X, , (6.104) | G ( X, ξ ) | k,X, + | G ( X, ξ ) | k,X, ≤ C | G ( X, ξ ) | k,X, ≤ C | K ( X, ξ ) | k,X, , (6.105) | G ( X, ξ ) | k,X, ≤ C | G ( X, ξ ) | k,X, ≤ C | K ( X, ξ ) | k,X, . (6.106) (cid:3) P Lemma . Consider the map P : M |k × M |k × c M : ,r → M ′ r defined in (6.10) , restricted to B ρ (1) × B ρ × c M : ,r ⊂ M |k × M |k × c M : ,r with theballs B ρ (1) and B ρ defined in terms of respective norms |k·k| k , i.e., B ρ (1) = { e I ∈ M |k : |k e I − k| k < ρ } and B ρ = { e J ∈ M |k : |k e J k| k < ρ } , and the target space M ′ r equipped with the norm k·k ( A ) k +1 ,r . There exists A = A ( L, d ) such that for any A ≥ A and ρ , ρ , and ˜ B such that (6.107) ρ ≤ / , ρ < (2 A d +2 ) − and ˜ B ≥ A d +3 the map P is smooth and, for any j , j ∈ N , satisfies the bounds (6.108) j ! 1 j ! k D j D j P ( e I, e J, e P )( ˙ e I, . . . , ˙ e I, ˙ e J, . . . , ˙ e J ) k ( A ) k +1 ,r ≤≤ |k ˙ e I k| j k (cid:0) A d +2 |k ˙ e J k| k (cid:1) j max (cid:16) k e P k ( A / , ˜ B ) k : k +1 ,r , (cid:17) , (6.109) j ! 1 j ! k D j D j D P ( e I, e J, e P )( ˙ e I, . . . , ˙ e I, ˙ e J, . . . , ˙ e J, ˙ e P ) k ( A ) k +1 ,r ≤≤ |k ˙ e I k| j k (cid:0) A d +2 |k ˙ e J k| k (cid:1) j k ˙ e P k ( A / , ˜ B ) k : k +1 ,r , (6.110) D j D j D j P = 0 for j ≥ . Proof. Since P is affine in the last argument, (6.110) is obvious and (6.109)follows from (6.108). Indeed since e P ( ∅ ) ≡ P can be written as(6.111) P ( e I, e J, e P ) = P ( e I, e J ) + P ( e I, e J, e P )with(6.112) P ( e I, e J )( U ) = X X ∈P ( U ) χ ( X , U ) e I U \ X e J X , (6.113) P ( e I, e J, e P ) = X X ,X ∈P ( U ) X ∩ X ∅ ,X = ∅ χ ( X ∪ X , U ) e I U \ ( X ∪ X ) e J X e P ( X )Since P is linear in P we have(6.114) D P ( e I, e J, e P )( ˙ e P ) = P ( e I, e J, ˙ e P ) = lim λ →∞ λ P ( e I, e J, λ e P )and an analogous identity holds for j ! 1 j ! D j D j D P . Thus (6.109) follows from(6.108).To prove (6.108) we first consider the case j = j = 0. Pick U ∈ P c k +1 . Takinginto account that k F ( U ) k k +1 ,U,r ≤ k F ( U ) k k : k +1 ,U,r , .6. THE MAP P and applying Lemma 5.1 (iib) we get(6.115) k P ( e I, e J, e P )( U ) k k +1 ,U,r ≤≤ X X ,X ∈P ( U ) X ∩ X ∅ ,X = ∅ χ ( X ∪ X , U ) |k e I k| | U \ ( X ∪ X ) | k |k e J k| | X | k k e P ( X ) k k : k +1 ,X ,r ≤ X X ,X ∈P ( U ) X ∩ X ∅ χ ( X ∪ X , U ) 2 | U \ ( X ∪ X ) | A − (1+2 d +2 ) | X | k e P k ( A / , ˜ B ) k : k +1 ,r Γ A / ( X ) − ˜ B −|C ( X ) | Now(6.116) Γ A / ( X ) ≥ (cid:0) A (cid:1) | X |− d |C ( X ) | and using that ˜ B ≥ A d +3 and 2 d +3 − d ≥ d +2 we get(6.117) k P ( e I, e J, e P )( U ) k k +1 ,U,r ≤≤ | U | X X ,X ∈P ( U ) X ∩ X ∅ ,X = ∅ χ ( X ∪ X , U ) A − (1+2 d +2 ) | X |−| X |− d +2 |C ( X ) | ) k e P k ( A / , ˜ B ) k : k +1 ,r . Now, we will rely on the combinatorial Lemma 6.16 from [ Bry09 ] stated in(F.2) in Lemma F.1,(6.118) | X | k ≥ (1+ α ( d )) | X | k +1 − (1+ α ( d ))2 d +1 |C ( X ) | with α ( d ) = d )(1+6 d ) . Applying this inequality with X = X ∪ X and using the trivial estimate C ( X ∪ X ) ≤ | X | + C ( X ), we get(6.119) (1 + 2 d +2 ) | X | k + | X | k + 2 d +2 |C ( X ) | ≥ (1 + α ( d )) | X ∪ X | k +1 and thus(6.120) k P ( e I, e J, e P )( U ) k k +1 ,U,r ≤ | U | k X X ,X ∈P ( U ) X ∩ X ∅ ,X = ∅ χ ( X ∪ X , U ) A − (1+ α ( d )) | X ∪ X | k +1 k e P k ( A / , ˜ B ) k : k +1 ,r . Similarly we obtain for P k P ( e I, e J )( U ) k k +1 ,U,r ≤ X X ∈P ( U ) χ ( X , U ) |k e I k| | U \ X | k |k e J k| | X | k (6.121) ≤ | U | X X ∈P ( U ) χ ( X , U ) A − (1+2 d +2 ) | X | Since α ( d ) ≤ ≤ d +2 and since | X | k ≥ | X | k +1 it is easy to combine the estimatesfor P and P . To prove (6.108) for j = j = 0 it thus suffices to show that(6.122) Γ A ( U ) 4 | U | k X X ,X ∈P ( U ) X ∩ X ∅ χ ( X ∪ X , U ) A − (1+ α ( d )) | X ∪ X | k +1 ≤ . for any U ∈ P c k +1 once(6.123) A ≥ A ( L, d ) = (12) L d (1+2 d )(1+6 d ) . If | U | k +1 ≤ d then Γ A ( U ) = 1 and we use | U | k = L d | U | k +1 as well as the factthat the sum in (6.122) has at most 3 | U | k ≤ L d d terms, each contributing at most A − ≤ A − d α ( d ) to bound the left hand side of (6.122) by(6.124) 4 (2 L ) d (2 L ) d A − ≤ (cid:16) (12) L d A − α ( d ) (cid:17) d ≤ . For | U | k +1 > d , there is no B ∈ P k such that U = B ∗ and as a result X ∪ X is not small and U = X ∪ X (cf. definition (4.69) of χ ( X ∪ X , U )). Hence, usingagain that the number of terms in the sum is bounded by 3 | U | k , we can bound theleft hand side of (6.122) by A | U | k +1 L d | U | k +1 A − (1+ α ( d )) | U | k +1 X X ,X ∈P ( U ) X ∩ X ∅ χ ( X ∪ X , U )(6.125) ≤ (12) L d | U | k +1 A − α ( d ) | U | k +1 ≤ L d A − α ( d ) ≤ j ! 1 j ! D j D j P ( e I, e J, e P )( U )( ˙ e I, . . . , ˙ e I, ˙ e J, . . . , ˙ e J )= X X ,X ∈P ( U ) X ∩ X ∅ ,X = ∅ χ ( X ∪ X , U ) X Y ∈P ( U \ ( X ∪ X )) , | Y | = j Y ∈P ( X ) , | Y | = j e I ( U \ ( X ∪ X )) \ Y ( ˙ e I ) Y e J X \ Y ( ˙ e J ) Y e P ( X )we proceed as above in (6.115) and (6.117) to get(6.127) j ! 1 j ! k D j D j P ( e I, e J, e P )( U )( ˙ e I, . . . , ˙ e I, ˙ e J, . . . , ˙ e J ) k k +1 ,U,r ≤≤ X X ,X ∈P ( U ) X ∩ X ∅ ,X = ∅ χ ( X ∪ X , U ) (cid:0) | U \ ( X ∪ X ) | j (cid:1) |k e I k| | U \ ( X ∪ X ) |− j k (cid:0) | X | j (cid:1) ×× |k e J k| | X |− j k k P ( X ) k k : k +1 ,X ,r |k ˙ e I k| j k |k ˙ e J k| j k ≤≤ X X ,X ∈P ( U ) X ∩ X ∅ ,X = ∅ χ ( X ∪ X , U )2 | U \ ( X ∪ X ) | | U \ ( X ∪ X ) |− j | X | × (2 A d +2 ) −| X | + j ( A ) −| X | +2 d |C ( X ) | A − d +3 |C ( X ) | k e P k ( A / , ˜ B ) k : k +1 ,r |k ˙ e I k| j k |k ˙ e J k| j k ≤≤ k e P k ( A / , ˜ B ) k : k +1 ,r |k ˙ e I k| j k (cid:0) A d +2 |k ˙ e J k| k (cid:1) j ×× | U | X X ,X ∈P ( U ) X ∩ X ∅ ,X = ∅ χ ( X ∪ X , U ) A − (1+2 d +2 ) | X |−| X |− d +2 |C ( X ) | ) . .7. PROOF OF PROPOSITION 4.6 73 Similarly we get(6.128) j ! 1 j ! k D j D j P ( e I, e J )( U )( ˙ e I, . . . , ˙ e I, ˙ e J, . . . , ˙ e J ) k k +1 ,U,r ≤≤ X X ∈P ( U ) χ ( X , U )2 | U \ X | | U \ X ) |− j | X | (2 A d +2 ) −| X | + j |k ˙ e I k| j k |k ˙ e J k| j k ≤ |k ˙ e I k| j k (cid:0) A d +2 |k ˙ e J k| k (cid:1) j | U | X X ∈P ( U ) χ ( X , U ) A − (1+2 d +2 ) | X | Now (6.108) follows as in the case j = j = 0 by using (6.119) and (6.122) as wellas the obvious estimates α ( d ) ≤ ≤ d +2 and | X | k ≥ | X | k +1 . (cid:3) Proposition 4.6 now follows from the estimates on the maps E, P , R , R , P and P and the chain rule, Theorem D.29, in connection with Remark D.30 whichprovides uniform control of the relevant derivatives. For the convenience of thereader we spell out the details. We first write S as a composition of five maps F , . . . , F and describe the scales of Banach spaces X ( i ) , i = 1 , . . . , 5, on whichthese maps are defined. Then we recursively identify neighbourhoods U ( i ) ⊂ X ( i ) such that F i ∈ e C m ( U ( i ) × B ) , i = 1 , . . . , , and verify that F i ( U ( i ) × B ) ⊂ U ( i − for i ≥ F i satisfies theassumptions of the chain rule Theorem D.29. Recall the definitions in Appendix Dand denote by ⋄ the composition defined by(6.129) (cid:0) F ⋄ G (cid:1) ( x , p ) := F ( G ( x , p ) , p ) . Define(6.130) ˜ B = A d +3 , B = 2 d ˜ B . In the following we will always assume(6.131) r ≥ m + 2 . We also assume that(6.132) A ≥ A ( L, d )where A ( L, d ) is the quantity in Lemma 6.10 and(6.133) h ≥ L κ ( d ) h with h = h ( d, ω )and κ ( d ) as in Lemma 5.1 (iv) (see (5.68)).Note that(6.134) S = F ⋄ F ⋄ F ⋄ F ⋄ F , where the maps F i , i = 1 , . . . , 5, and the scales of Banach spaces are given by(6.135) F : X (1) × B → X (0) , F ( K , K , K , q ) = P ( K , K , K ) , with(6.136) X (1) n = M |k × ( c M : ,r − m +2 n , k·k ( A / , ˜ B ) k : k +1 ,r − m +2 n ) X (0) n = ( M ′ r − m +2 n , k·k ( A ) k +1 ,r − m +2 n ) ,B = { q ∈ R d × d sym : k q k < } ;and(6.137) F : X (2) × B → X (1) , F ( H, K, q ) := ( E ( H ) , − E ( H ) , R ( K, q )) , with(6.138) X (2) n = ( M , k·k k, ) × ( c M r − m +2 n , k·k ( A / , B ) k,r − m +2 n );and(6.139) F : X (3) → X (2) , F ( H, K ) := ( H, P ( K )) , with(6.140) X (3) n = ( M , k·k k, ) × ( M r − m +2 n , k·k ( A / k,r − m +2 n )(6.141) F : X (4) × B → X (3) , F ( H, e K, K, q ) := ( R ( H, K, q ) , P ( e K, K )) , with(6.142) X (4) n = ( M , k·k k, ) × M |k × ( M r − m +2 n , k·k ( A ) k,r )and(6.143) F : X (5) × B → X (4) , F ( H, K ) := ( H, E ( H ) , K ) , with(6.144) X (5) n = ( M , k·k k, ) × ( M r − m +2 n , k·k ( A ) k,r − m +2 n ) . Let(6.145) U (1) = B ρ (1) × B ρ × c M : ,r ⊂ X (1) m with ρ ≤ , ρ < (cid:0) A d +2 (cid:1) − . Then by Lemma 6.10 we have(6.146) F ∈ e C m ( U (1) × B , X (0) ) , and the derivatives of F satisfy the assumptions of the chain rule, Theorem D.29.Let C . C . ≥ ρ = 1 C . { ρ , ρ } = ρ C . . Then H ∈ B ρ implies that E ( H ) − ∈ B ρ ∩ B ρ ⊂ M |k . Thus the choice U (2) := B ρ × c M r yields(6.148) F ( U (2) × B ) ⊂ U (1) . .7. PROOF OF PROPOSITION 4.6 75 Moreover by Lemma 6.1 and Lemma 6.5 the map F : U (2) × B → X (1) m satisfiesthe assumptions of the chain rule, Theorem D.29.Let ρ := (2 B ) − , U (3) = B ρ × B ρ Then(6.149) F ( U (3) × B ) ⊂ U (2) and by Lemma 6.4 the map F is a smooth map on U (3) and on U (3) satisfies the as-sumptions of the chain rule Theorem D.29. Note that we are applying Lemma D.32for those maps which do not depend on q like F , F and F .We have ρ ≤ 1. Let C . ρ = ρ C . , ρ = ρ A , ρ = min n ρ C . , ρ A d o . Then it follows from (6.34) in Lemma 6.3 and Lemma 6.7 (with r = r ) that(6.151) F ( B ρ × B ρ (1) × B ρ × B ) ⊂ B ρ × B ρ = U (3) . Set U (4) := B ρ × B ρ (1) × B ρ . Then F (4) : U (4) × B → X (3) m satisfies the assump-tions of the chain rule.Finally set(6.152) ρ = ρ C . , ρ = ρ , and U (5) = B ρ × B ρ . Then F ( U (5) × B ) ⊂ U (4) and F : U (5) × B → X (4) m satisfies the assumptionsof the chain rule. Now an application of the chain rule, Theorem D.29, shows thatthe conclusions of Proposition 4.6 hold with ρ = min { ρ , ρ } . (cid:3) HAPTER 7 Linearization of the Renormalization Map Here we prove Proposition 4.7 summarizing the properties of the linearization(4.80) of the maps T k at the fixed point ( H k , K k ) = (0 , 0) guaranteeing that H k and K k are the relevant and irrelevant variables, respectively. First, we prove thecontraction property of the operator C ( q ) in Section 7.2. We finish the proof ofProposition 4.7 in Section 7.2 with the bounds on the operators A ( q ) − and B ( q ) . C ( q ) Lemma . Let θ ∈ ( , ) and ω ≥ d d +1 +1) . Consider the constant h = h ( d, ω ) and κ ( d ) chosen from Lemma 5.1 and let L ≥ d + 1 , h ≥ L κ ( d ) h ( d, ω ) .There exists A = A ( d, L ) such that (7.1) k C ( q ) k ( A ) r = sup k K k ( A ) k,r ≤ k C ( q ) K k ( A ) k +1 ,r ≤ θ. for any k q k ≤ , any k = 1 , . . . , N , r = 1 , . . . , r , and any A ≥ A . Proof. Let us begin by evaluating the large set term: the last term on the righthand side of (4.83). Lemma . Let L ≥ d + 1 and ω ≥ √ . Whenever h ≥ L κ ( d ) h ,and A such that A − α α ≤ δ ( d, L ) with α from Lemma F.1 and δ ( d, L ) fromLemma F.2, then (7.2) k F k ( A ) k +1 ,r ≤ θ k K k ( A ) k,r for any K ∈ M ( P k , X ) . Here, the function F ∈ M ( P k +1 , X ) is defined by (7.3) F ( U, ϕ ) = X X ∈P c k \S k X = U Z X K ( X, ϕ + ξ ) µ k +1 (d ξ ) . Proof. Considering, for any X ⊂ U , the function ( R k +1 K )( X, ϕ ) and its norm | ( R k +1 K )( X, ϕ ) | k +1 ,U,r as defined by (4.25), we have(7.4) sup ϕ | ( R k +1 K )( X, ϕ ) | k +1 ,U,r w − Uk +1 ≤ sup ϕ | ( R k +1 K )( X, ϕ ) | k +1 ,X,r w − Xk : k +1 . To see it, we just notice that, as in (5.4) in the proof of Lemma 5.1, one has(7.5) | ( R k +1 K )( X, ϕ ) | k +1 ,U,r ≤ | ( R k +1 K )( X, ϕ ) | k +1 ,X,r and that(7.6) w − Uk +1 ( ϕ ) ≤ w − Xk : k +1 . 778 7. LINEARIZATION OF THE RENORMALIZATION MAP The last inequality amounts to(7.7) X x ∈ X (cid:0) (2 d ω − g k : k +1 ,x ( ϕ ) + ωG k,x ( ϕ ) (cid:1) + 3 L k X x ∈ ∂X G k,x ( ϕ ) ≤≤ X x ∈ U ω (cid:0) d g k +1 ,x ( ϕ ) + G k +1 ,x ( ϕ ) (cid:1) + L k +1 X x ∈ ∂U G k +1 ,x ( ϕ ) . This is clearly valid since g k : k +1 ,x ( ϕ ) ≤ g k +1 ,x ( ϕ ), G k,x ( ϕ ) ≤ G k +1 ,x ( ϕ ), and any x ∈ ∂X \ ∂U is necessarily contained in ∂B for some B ∈ B k ( U \ X ) and, in viewof (5.16), for each such B one has(7.8) 3 L k X x ∈ ∂B G k,x ( ϕ ) ≤ X x ∈ B ω (cid:0) d g k +1 ,x ( ϕ ) + G k +1 ,x ( ϕ ) (cid:1) once ω ≥ c + 1.Combining now (7.4) with the bound from Lemma 5.1 (iv), we get(7.9) Γ k +1 ,A ( U ) k F ( U ) k k +1 ,U,r ≤ A | U | k +1 X X ∈P c k \S k X = U | X | k k K ( X ) k k,X,r ≤≤ k K k ( A ) k,r A | U | k +1 X X ∈P c k \S k X = U ( A ) −| X | k ≤ k K k ( A ) k,r X X ∈P c k \S k X = U (2 A − α α ) | X | k ≤ θ k K k ( A ) k,r . Here, in the last two inequalities, we first used | X | k ≥ (1 + 2 α ( d )) | X | k +1 for any X contributing to the sum (see [ Bry09 , Lemma 6.15]; (F.1) in Lemma F.1) and thenapplied Lemma F.2 assuming that 2 A − α α ≤ θ δ ( d, L ). (cid:3) Turning to the first term on the right hand side of (4.83), we have: Lemma . Let L ≥ , ω ≥ d d +1 + 1) , h ≥ L κ ( d ) h , and K ∈ M ( P k , X ) with G ∈ M ( P k +1 , X ) defined by (7.10) G ( U, ϕ ) = X B ∈B k ( U ) B ∗ = U (cid:0) − Π (cid:1) X X ∈S k,X ⊃ B | X | k ( R k +1 K )( X, ϕ ) . Then (7.11) k G k ( A ) k +1 ,r ≤ d +2 d (3 d − d (cid:0) L − d + 2 d +3 L d − + 9 L − (cid:1) k K k ( A ) k,r for any A > . Remark . Notice that (7.11) is used later only for d ≤ 3. Our method canbe extended also to include higher dimension when employing additional higherorder terms to estimate the projection of the second Taylor polynomial. ⋄ Proof. Notice first that the sum vanishes unless U ∈ S k +1 and, necessarily, forany contributing X , one has X ⊂ U and X ∗ ⊂ U ∗ . As a result, the norms in (7.11)contain only the contributions of small sets and do not depend on A according to thedefinition of the factor Γ j, A ( X ), j = k, k + 1. Considering R ∈ M ∗ ( B k , X ) definedby R ( B, ϕ ) = P X ∈S kX ⊃ B | X | k ( R k +1 K )( X, ϕ ) and replacing the operator 1 − Π by(1 − T ) + ( T − Π ), we split G ( U, ϕ ) into two terms,(7.12) G ( U, ϕ ) = X B ∈B k ( U ) B ∗ = U (1 − T ) R ( B, ϕ ) .1. CONTRACTIVITY OF OPERATOR C ( q ) and(7.13) G ( U, ϕ ) = X B ∈B k ( U ) B ∗ = U ( T − Π ) R ( B, ϕ ) , and evaluate them separately in Lemma 7.6 and Lemma 7.7.First, however, considering the norm | F ( X, ϕ ) | j,X,r , j = k, k + 1, as defined in(4.27) for any F ∈ M ( P k , X ) with X ∈ P k and ϕ ∈ X , we prove the following. Lemma . Let F ∈ M ( P k , X ) , X ∈ P k , r = 1 , . . . , r , and j = k, k + 1 .Then (7.14) | F ( X, ϕ ) − T F ( X, ϕ ) | j,X,r ≤ (1 + | ϕ | j,X ) sup t ∈ (0 , r X s =3 s ! | D s F ( X, tϕ ) | j,X . Proof. Cf. [ Bry09 , Lemma 6.8]. Introducing the shorthands f ( ϕ ) = (1 − T ) F ( X, ϕ )and f s ( ϕ ) = D s F ( X, ϕ )( ˙ ϕ, . . . , ˙ ϕ )for any s ≥ 1, we express the terms contributing to the left hand side of (7.14) withthe help of the integral form of the Taylor polynomial remainder,(7.15) f ( ϕ ) = Z (1 − t ) D F ( X, tϕ )( ϕ, ϕ, ϕ ) d t, (7.16) Df ( ϕ )( ˙ ϕ ) = f ( ϕ ) − f (0) − Df (0)( ϕ ) = Z (1 − t ) D f ( tϕ )( ϕ, ϕ ) d t == Z (1 − t ) D F ( X, tϕ )( ˙ ϕ, ϕ, ϕ ) d t, (7.17) 12 D f ( ϕ )( ˙ ϕ, ˙ ϕ ) = 12 (cid:0) f ( ϕ ) − f (0) (cid:1) == 12 Z Df ( tϕ )( ϕ ) d t = Z D F ( X, tϕ )( ˙ ϕ, ˙ ϕ, ϕ ) d t, and, for s ≥ s ! D s f ( ϕ )( ˙ ϕ, . . . , ˙ ϕ ) = 1 s ! D s F ( X, ϕ )( ˙ ϕ, . . . , ˙ ϕ ) . Summing all the right hand sides above and using the bound(7.19) | D s + m F ( X, tϕ )( ˙ ϕ, . . . , ˙ ϕ, ϕ, . . . , ϕ ) | ≤ | D s + m F ( X, tϕ ) | j,X | ˙ ϕ | sj,X | ϕ | mj,X , as well as the fact that(7.20) | ϕ | j,X Z (1 − t ) t + | ϕ | j,X Z (1 − t ) d t + 12 | ϕ | j,X + 13! = 13! (1 + | ϕ | j,X ) , we get the seeked result. (cid:3) Lemma . Let K ∈ M ( S k , X ) , X ∈ S k , B ∈ B k ( X ) , and U = B ∗ , andassume that L ≥ , ω ≥ d d +1 + 1) , and h ≥ L κ ( d ) h . Then (7.21)sup ϕ | ( R k +1 K )( X, ϕ ) − T ( R k +1 K )( X, ϕ ) | k +1 ,X,r w − Uk +1 ( ϕ ) ≤ L − d | X | k k K ( X ) k k,X,r . For G defined in (7.12) we have (7.22) k G ( U ) k k +1 ,U,r ≤ d +2 d (3 d − d L − d k K k ( A ) k,r . Proof. Lemma 7.5 yields(7.23) | ( R k +1 K )( X, ϕ ) − T ( R k +1 K )( X, ϕ ) | k +1 ,X,r ≤≤ (1 + | ϕ | k +1 ,X ) sup t ∈ (0 , r X s =3 s ! | D s ( R k +1 K )( X, tϕ ) | k +1 ,X for any ϕ ∈ X . Interchanging differentiation and integration, we get(7.24) r X s =3 s ! | D s ( R k +1 K )( X, tϕ ) | k +1 ,X ≤≤ r X s =3 s ! sup ˙ ϕ =0 Z X µ k +1 (d ξ ) (cid:12)(cid:12)(cid:12) D s K ( X, tϕ + ξ )( ˙ ϕ, . . . , ˙ ϕ ) | ˙ ϕ | sk +1 ,X (cid:12)(cid:12)(cid:12) == r X s =3 s ! sup ˙ ϕ =0 Z X µ k +1 (d ξ ) (cid:12)(cid:12)(cid:12) D s K ( X, tϕ + ξ )( ˙ ϕ, . . . , ˙ ϕ ) | ˙ ϕ | sk,X | ˙ ϕ | sk,X | ˙ ϕ | sk +1 ,X (cid:12)(cid:12)(cid:12) ≤≤ L − d Z X µ k +1 (d ξ ) | K ( X, tϕ + ξ ) | k,X,r . In the last inequality we used the bound (5.21). Next, we apply | K ( X, tϕ + ξ ) | k,X,r ≤ k K ( X ) k k,X,r w Xk ( tϕ + ξ )and (5.25), to get(7.25) r X s =3 s ! | D s ( R k +1 K )( X, tϕ ) | k +1 ,X ≤ | X | k L − d k K ( X ) k k,X,r w Xk : k +1 ( ϕ ) w Uk +1 ( ϕ ) w Uk +1 ( ϕ ) . Here we also used the fact that w Xk : k +1 ( tϕ ) is monotone in t .Bounding (1 + | ϕ | k +1 ,X ) with the help of(7.26) (1 + u ) ≤ u (proven by showing that min u ≥ u (1+ u ) ≥ ), we would like to show that(7.27) | ϕ | k +1 ,X ≤ log w Uk +1 ( ϕ ) w Xk : k +1 ( ϕ ) . .1. CONTRACTIVITY OF OPERATOR C ( q ) Notice, first, that(7.28)log w Uk +1 ( ϕ ) w Xk : k +1 ( ϕ ) ≥ X x ∈ U \ X (cid:0) (2 d ω − g k +1 ,x ( ϕ ) + ωG k +1 ,x ( ϕ ) (cid:1) + X x ∈ U g k : k +1 ,x ( ϕ )++ L k ( L − X x ∈ ∂U G k +1 ,x ( ϕ ) − L k X x ∈ ∂X \ ∂U G k,x ( ϕ ) ≥≥ X x ∈ U \ X (2 d ω − g k +1 ,x ( ϕ ) + L k ( L − X x ∈ ∂U G k +1 ,x ( ϕ ) . To verify the last inequality, we show that(7.29) 3 L k X x ∈ ∂X \ ∂U G k,x ( ϕ ) ≤ X x ∈ U g k : k +1 ,x ( ϕ ) + X x ∈ U \ X ωG k +1 ,x ( ϕ )in analogy with (5.15). Indeed, arguing that any x ∈ ∂X \ ∂U is contained in ∂B for B ∈ B k ( U \ X ), and applying again Proposition B.5 (a), we have(7.30) h L k X x ∈ ∂B G k,x ( ϕ ) ≤≤ c (cid:0)X x ∈ B |∇ ϕ ( x ) | + L k X x ∈ U ( B ) |∇ ϕ ( x ) | (cid:1) + L k X x ∈ ∂B X s =2 L (2 s − k |∇ s ϕ ( x ) | ≤≤ h c X x ∈ B G k,x ( ϕ ) + h c L k X x ∈ ∂B L − g k : k +1 ,z ( ϕ ) , where z is any point z ∈ B . Using | ∂B | ≤ d L ( d − k , we get the seeked bound once ω ≥ √ L ≥ c ≤ ω and 6 c L − ≤ | ϕ | k +1 ,X ≤ | ϕ | k +1 ,U , it suffices to show that(7.31) | ϕ | k +1 ,U ≤ X x ∈ U \ X (2 d ω − g k +1 ,x ( ϕ ) + L k ( L − X x ∈ ∂U G k +1 ,x ( ϕ ) . Clearly,(7.32) h | ϕ | k +1 ,U ≤ X ≤ s ≤ L ( k +1)( d − s ) max x ∈ U ∗ |∇ s ϕ ( x ) | Applying Lemma B.7, we get(7.33) L ( k +1) d max x ∈ U ∗ |∇ ϕ ( x ) | ≤ L ( k +1) d | ∂U | X x ∈ ∂U |∇ ϕ ( x ) | +2 L ( k +1) d (diam U ∗ ) max x ∈ U ∗ |∇ ϕ ( x ) | . Using that | ∂U | ≥ dL ( k +1)( d − , the first term above is covered by the second termon the right hand side of (7.31) once L ≥ L ( k +1) d | ∂U | ≤ L ( k +1) d dL ( k +1)( d − = 1 d L k +1 ≤ L k ( L − . Taking into account that diam U ∗ ≤ d d L k +1 (here we use the fact that U is nec-essarily contained in a block of the side 2 L k +1 ), the second term is bounded by d d +1 L ( k +1)( d +2) max x ∈ U ∗ |∇ ϕ ( x ) | and will be treated together with the re-maining terms max x ∈ U ∗ |∇ s ϕ ( x ) | , s = 2 , 3, contained in | ϕ | k +1 ,U . Using the fact that the number of ( k + 1)-blocks in U is at most 2 d , we get(7.35) max x ∈ U ∗ |∇ s ϕ ( x ) | ≤ d X B ∈B k +1 ( U ) max x ∈ B ∗ |∇ s ϕ ( x ) | . This yields(7.36) ( d d +1 L ( k +1)( d +2) + L ( k +1)( d +2) ) max x ∈ U ∗ |∇ ϕ ( x ) | ≤≤ d ( d d +1 + 1) L ( k +1)( d +2) X B ∈B k +1 ( U ) max x ∈ B ∗ |∇ ϕ ( x ) | . and(7.37) L ( k +1)( d +4) max x ∈ U ∗ |∇ ϕ ( x ) | ≤ d L ( k +1)( d +4) X B ∈B k +1 ( U ) max x ∈ B ∗ |∇ ϕ ( x ) | . Each of the terms on the right hand sides will be bounded by the correspondingterm in(7.38) h X x ∈ B \ X (2 d ω − g k +1 ,x ( ϕ ) = (2 d ω − X x ∈ B \ X X s =2 L (2 s − k +1) sup y ∈ B ∗ x |∇ s ϕ ( y ) | , Indeed, observing that g k +1 ,x ( ϕ ) is constant over each ( k + 1)-block B ⊂ U , andthe volume of B \ X is at least L kd ( L d − d ) = L ( k +1) d (1 − ( L ) d ) since the numberof k -blocks in X is at most 2 d , while B consists of L d of them, we need(7.39) 2 d ( d d +1 + 1) L ( k +1)( d +2) ≤ (2 d ω − L ( k +1) d (1 − ( L ) d ) L k +1) and(7.40) 2 d L ( k +1)( d +4) ≤ (2 d ω − L ( k +1) d (1 − ( L ) d ) L k +1) . These conditions are satisfied once ω ≥ d d +1 + 1).In summary, combining (7.25), (7.26), and (7.27), we have(7.41) (1 + | ϕ | k +1 ,X ) r X s =3 s ! | D s ( R k +1 K )( X, tϕ ) | k +1 ,X ≤≤ L − d | X | k k K ( X ) k k,X,r w Uk +1 ( ϕ ) . for any ϕ ∈ X and any t ∈ (0 , |B k ( U ) | ≤ (2 L ) d and the obvious bound |{ X ∈ S k | X ⊃ B }| ≤ (3 d − d , to get(7.42) k G ( U ) k k +1 ,U,r ≤ L − d X B ∈B k ( U ) B ∗ = U X X ∈S kX ⊃ B | X | k | X | k k K ( X ) k k,X,r ≤≤ L − d (2 L ) d (3 d − d k K k ( A ) k,r d ≤ d +2 d (3 d − d L − d k K k ( A ) k,r . (cid:3) .1. CONTRACTIVITY OF OPERATOR C ( q ) Lemma . Let K ∈ M ( S k , X ) , U = B ∗ , and assume that L ≥ and ω ≥ d d +1 + 1) . For G defined in (7.13) we have (7.43) k G ( U ) k k +1 ,U,r ≤≤ d + d +1 (3 d − d (cid:0) (2 d +2 − L d − + (8 L − + 2 L − ) (cid:1) k K k k,r . Recall that G ( U, ϕ ) = P B ∈B k ( U ) B ∗ = U ( T − Π ) R ( B, ϕ ) with R ∈ M ∗ ( B k , X ) de-fined by R ( B, ϕ ) = P X ∈S kX ⊃ B | X | k ( R k +1 K )( X, ϕ ). The polynomial Π R ( B, ϕ ) = λ | B | + ℓ ( ϕ ) + Q ( ϕ, ϕ ) is characterised by taking a unique linear function ℓ ( ϕ ) of theform (4.19), ℓ ( ϕ ) = P x ∈ ( B ∗ ) ∗ (cid:2)P di =1 a i ∇ i ϕ ( x ) + P di,j =1 c i,j ∇ i ∇ j ϕ ( x ) (cid:3) , that agreeswith DR ( B, ϕ ) on all quadratic functions ϕ on ( B ∗ ) ∗ and a unique quadraticfunction Q ( ϕ, ϕ ) of the form (4.20), Q ( ϕ, ϕ ) = P x ∈ ( B ∗ ) ∗ P di,j =1 d i,j ∇ i ϕ ( x ) ∇ j ϕ ( x ),that agrees with D R ( B, ϕ, ϕ ) on all affine functions ϕ on ( B ∗ ) ∗ .In view of the definition of the map R k +1 we can write R ( B, ϕ ) = Z X µ k +1 (d ξ ) R ξ ( B, ϕ )with R ξ ( B, ϕ ) = X X ∈S kX ⊃ B | X | k K ( X, ξ + ϕ ) . Observing that D ( R k +1 K )( X, ϕ ) = Z X µ k +1 (d ξ ) DK ( X, ξ )( ϕ ) ,D ( R k +1 K )( X, ϕ, ϕ ) = Z X µ k +1 (d ξ ) D K ( X, ξ )( ϕ, ϕ ) , and introducing, similarly as above, Π R ξ ( B, ϕ ) = λ ξ | B | + ℓ ξ ( ϕ ) + Q ξ ( ϕ, ϕ ), theunicity implies that ℓ ( ϕ ) = R X µ k +1 (d ξ ) ℓ ξ ( ϕ ) and Q ( ϕ, ϕ ) = R X µ k +1 (d ξ ) Q ξ ( ϕ, ϕ ).Given that G ( B, ϕ ) = ( T − Π ) R ( B, ϕ ) is a polynomial of second order, wehave | G ( B, ϕ ) | k +1 ,U,r = | G ( B, ϕ ) | k +1 ,U, . In a preparation for the evaluation ofthis norm, we first evaluate separately the absolute value of the linear and quadraticterms P ( ϕ ) and P ( ϕ ) in G ( B, ϕ ).Observing that for any affine function ϕ and any quadratic function ϕ on( B ∗ ) ∗ we have P ( ϕ − ϕ − ϕ ) = P ( ϕ ), we get(7.44) (cid:12)(cid:12) P ( ϕ ) (cid:12)(cid:12) = (cid:12)(cid:12)Z X µ k +1 (d ξ ) (cid:0) DR ξ ( B, ϕ − ϕ − ϕ ) − ℓ ξ ( ϕ − ϕ − ϕ ) (cid:1)(cid:12)(cid:12) ≤≤ (2 d +2 − X X ∈S kX ⊃ B | X | k k K ( X ) k k,X,r | ϕ − ϕ − ϕ | k,B ∗ Z X µ k +1 (d ξ ) w Xk ( ξ ) ≤≤ d (3 d − d (2 d +2 − k K k k,r | ϕ − ϕ − ϕ | k,B ∗ . Here, we first used the inequalities(7.45) | ℓ ξ ( ϕ ) | ≤ (2 d +2 − X X ∈S kX ⊃ B | X | k | K ( X, ξ ) | k,X,r | ϕ | k,B ∗ and(7.46) | DR ξ ( B, ϕ ) | ≤ X X ∈S kX ⊃ B | X | k | K ( X, ξ ) | k,X,r | ϕ | k,X combined with the bounds | K ( X, ξ ) | k,X,r ≤ k K ( X ) k k,X,r w Xk ( ξ ) and | ϕ | k,X ≤ | ϕ | k,B ∗ , and then the bounds R X µ k +1 (d ξ ) w Xk ( ξ ) ≤ | X | k , and, as in (7.42), |{ X ∈S k | X ⊃ B }| ≤ (3 d − d . To verify (7.45), we first observe that ℓ ξ ( ϕ ) = P di =1 a i ( ξ ) s i + P di,j =1 c i,j ( ξ ) t i,j where s i = s i ( ϕ ) = P x ∈ ( B ∗ ) ∗ ∇ i ϕ ( x ) and t i,j = t i,j ( ϕ ) = P x ∈ ( B ∗ ) ∗ ∇ i ∇ j ϕ ( x ). The same values of “average slopes” s = { s i } and t = { t i,j } are obtained with the quadratic function(7.47) ϕ s , t ( x ) = L − dk (2 d +2 − − d X i (cid:0) s i − X j ( t i,j + t j,i ) x j (cid:1) x i + L − dk (2 d +2 − − d X i,j t i,j x i x j , where x j = L − dk (2 d +2 − − d P y ∈ B y j (notice that ( B ∗ ) ∗ contains (2 d +2 − d k -blocks). Further, observe that(7.48) h | ϕ s , t | k,X = max (cid:16) L dk max x ∈ X ∗ |∇ ϕ s , t ( x ) | , L dk + k max x ∈ X ∗ |∇ ϕ s , t ( x ) | (cid:1) ≤≤ L − dk (2 d +2 − − d max (cid:16) | s | + 2 | t | L k (2 d +2 − , L k | t | (cid:17) == L − dk (2 d +2 − − d | s | + L − dk + k (2 d +2 − − d +1 | t | ≤ (1 + 2 d +2 − h | ϕ | k,B ∗ . Here, the last inequality, valid for any ϕ such that s i ( ϕ ) = s i and t i,j ( ϕ ) = t i,j ,is implied by obvious bounds max x ∈ ( B ∗ ) ∗ |∇ i ϕ ( x ) | ≥ L − dk (2 d +2 − − d | s i | andmax x ∈ ( B ∗ ) ∗ |∇ i ∇ j ϕ ( x ) | ≥ L − dk (2 d +2 − − d | t i,j | .Now, for the quadratic function ϕ s , t we have ℓ ξ ( ϕ s , t ) = DR ξ ( B, ϕ s , t ). As aresult,(7.49) | ℓ ξ ( ϕ ) | = | ℓ ξ ( ϕ s , t ) | ≤≤ X X ∈S kX ⊃ B | X | k | DK ( X, ξ )( ϕ s , t ) | ≤ X X ∈S kX ⊃ B | X | k | K ( X, ξ ) | k,X,r | ϕ s , t | k,X ≤≤ (2 d +2 − X X ∈S kX ⊃ B | X | k | K ( X, ξ ) | k,X,r | ϕ | k,B ∗ . Here, the last inequality, valid for any ϕ such that s i ( ϕ ) = s i and t i,j ( ϕ ) = t i,j ,is implied by obvious bounds max x ∈ ( B ∗ ) ∗ |∇ i ϕ ( x ) | ≥ L − dk (2 d +2 − − d | s i | andmax x ∈ ( B ∗ ) ∗ |∇ i ∇ j ϕ ( x ) | ≥ L − dk (2 d +2 − − d | t i,j | .Choosing now, for any fixed ϕ , the functions ϕ and ϕ as an optimal approx-imation in accordance with the Poincar´e inequalities,(7.50) inf ϕ affine | ϕ − ϕ | k,B ∗ ≤ h L k ( d +1) sup x ∈ ( B ∗ ) ∗ |∇ ϕ ( x ) | ≤ L − ( d +1) | ϕ | k +1 ,B ∗ and(7.51)inf ϕ affine, ϕ quadratic | ϕ − ϕ − ϕ | k,B ∗ ≤ h L k ( d +2) sup x ∈ ( B ∗ ) ∗ |∇ ϕ ( x ) | ≤ L − ( d +2) | ϕ | k +1 ,B ∗ , .1. CONTRACTIVITY OF OPERATOR C ( q ) we get(7.52) (cid:12)(cid:12) P ( ϕ ) (cid:12)(cid:12) ≤ L − ( d +2) d (3 d − d (2 d +2 − k K k k,r | ϕ | k +1 ,B ∗ . Similarly for the quadratic part. First, we prove the bound(7.53) | P ( ϕ, ϕ ) | ≤ d +1 (3 d − d k K k k,r | ϕ | k,B ∗ . While deriving it, the bound (7.45) is replaced by(7.54) | Q ξ ( ϕ, ϕ ) | ≤ X X ∈S kX ⊃ B | X | k | K ( X, ξ ) | k,X,r | ϕ | k,B ∗ . For its proof we consider the linear function(7.55) ϕ s ( x ) = L − dk (2 d +2 − − d X i s i x i with the slope s i = s i ( ϕ ) and(7.56) h | ϕ s | k,X == L dk max x ∈ X ∗ |∇ ϕ s ( x ) | ≤ L − dk (2 d +2 − − d | s | = L − dk (2 d +2 − − d | s | ≤ h | ϕ | k,B ∗ yielding(7.57) | Q ξ ( ϕ, ϕ ) | = | Q ξ ( ϕ s , ϕ s ) | ≤ X X ∈S kX ⊃ B | X | k | D K ( X, ξ )( ϕ s , ϕ s ) | ≤≤ X X ∈S kX ⊃ B | X | k | K ( X, ξ ) | k,X,r | ϕ s | k,X ≤ X X ∈S kX ⊃ B | X | k | K ( X, ξ ) | k,X,r | ϕ | k,B ∗ . Validity of (7.53) for all ϕ , implies | P ( ϕ, ψ ) | ≤ d +2 (3 d − d k K k k,r | ϕ | k,B ∗ | ψ | k,B ∗ for all ϕ and ψ . Taking now into account that P ( ϕ , ϕ ) = 0 for any affine function ϕ , we rewrite P ( ϕ, ϕ ) = 2 P ( ϕ, ϕ − ϕ ) − P ( ϕ − ϕ , ϕ − ϕ ) to get(7.58) | P ( ϕ, ϕ ) | ≤ d +1 (3 d − d k K k ( A ) k,r | ϕ − ϕ | k,B ∗ (4 | ϕ | k,B ∗ + | ϕ − ϕ | k,B ∗ ) . Applying further (7.50), we get(7.59) | P ( ϕ, ϕ ) | ≤ (cid:0) L − ( d +1) + L − ( d +2) (cid:1) d +1 (3 d − d k K k ( A ) k,r | ϕ | k +1 ,B ∗ . Finally, combining (7.52) and (7.59), we get(7.60) | (cid:0) T − Π (cid:1) R ( B, ϕ ) | ≤≤ d (3 d − d (cid:0) (2 d +2 − L − ( d +2) +(8 L − ( d +1) +2 L − ( d +2) ) | ϕ | k +1 ,B ∗ (cid:1) | ϕ | k +1 ,B ∗ k K k ( A ) k,r . For the first and second the derivatives, we first notice that(7.61) D (cid:0) P ( ϕ ) + P ( ϕ, ϕ ) (cid:1) ( ˙ ϕ ) = P ( ˙ ϕ ) + 2 P ( ϕ, ˙ ϕ )and(7.62) D (cid:0) P ( ϕ ) + P ( ϕ, ϕ ) (cid:1) ( ˙ ϕ, ˙ ϕ ) = 2 P ( ˙ ϕ, ˙ ϕ ) yielding with the help of (7.52) and (7.59)(7.63) (cid:12)(cid:12) D (cid:0) P ( ϕ ) + P ( ϕ, ϕ ) (cid:1)(cid:12)(cid:12) k +1 ,B ∗ ≤≤ d (3 d − d (cid:0) (2 d +2 − L − ( d +2) + (16 L − ( d +1) + 4 L − ( d +2) ) | ϕ | k +1 ,B ∗ (cid:1) k K k ( A ) k,r and, using again (7.59),(7.64) (cid:12)(cid:12) D (cid:0) P ( ϕ ) + P ( ϕ, ϕ ) (cid:1)(cid:12)(cid:12) k +1 ,B ∗ ≤ d (3 d − d (8 L − ( d +1) + 2 L − ( d +2) ) k K k ( A ) k,r . Combining last two inequalities with (7.60), we get(7.65) | (cid:0) T − Π (cid:1) R ( B, ϕ ) | k +1 ,B ∗ ,r ≤ d (3 d − d (cid:0) (2 d +2 − L − ( d +2) ++ (8 L − ( d +1) + 2 L − ( d +2) )(1 + | ϕ | k +1 ,B ∗ ) (cid:1) (1 + | ϕ | k +1 ,B ∗ ) k K k ( A ) k,r . With (1 + u ) ≤ u and (7.27), we get(7.66) k G ( U ) k k +1 ,U,r ≤≤ d +1 (3 d − d (2 L ) d (cid:0) (2 d +2 − L − ( d +2) + (8 L − ( d +1) + 2 L − ( d +2) ) (cid:1) k K k ( A ) k,r yielding the sought bound. (cid:3) The proof of Lemma 7.1 is the finished by combining the claims of Lemma 7.2and Lemma 7.3. (cid:3) A ( q ) − and B ( q ) The bounds on operators A − and B are rather straightforward. Lemma . Let θ ∈ ( , ) and ω ≥ d d +1 + 1) . Consider the constant h = h ( d, ω ) , κ ( d ) , A = A ( d, L ) as chosen from Lemma 7.1. Then there exists L ( d ) such that (7.67) k A ( q ) − k ≤ √ θ and there exists M = M ( d ) such that (7.68) k B ( q ) k r ;0 ≤ M L d for any k q k ≤ , any N ∈ N , k = 1 , . . . , N , r = 1 , . . . , r , and any L ≥ L , h ≥ L κ h , and A ≥ A . Proof. When expressed in the coordinates ˙ λ, ˙ a, ˙ c , ˙ d of ˙ H , the linear map A accordingto (4.81) keeps ˙ a, ˙ c , and ˙ d unchanged and only shifts ˙ λ by12 X x ∈ B d X i,j =1 ˙ d i,j ∇ i ∇ ∗ j C ( q ) k +1 (0) . .2. BOUNDS ON THE OPERATORS A ( q ) − AND B ( q ) Hence, A − only makes the opposite shift and thus(7.69) k A − ˙ H k k, == L dk | ˙ λ | + L dk h d X i =1 | ˙ a i | + L ( d − k h d X i,j =1 | ˙ c i,j | + h d X i,j =1 | ˙ d i,j | + L dk d X i,j =1 | ˙ d i,j | (cid:12)(cid:12) ∇ i ∇ ∗ j C ( q ) k +1 (0) (cid:12)(cid:12) . Using(7.70) 12 d X i,j =1 | ˙ d i,j | ≤ h k ˙ H k k, , we get k A − ˙ H k k, ≤ (1 + c , L η ( d ) h − ) k ˙ H k k +1 , using that max di,j =1 (cid:12)(cid:12) ∇ i ∇ ∗ j C ( q ) k +1 (0) (cid:12)(cid:12) ≤ c , L − kd L η ( d ) according to Proposition 4.1.Given that h ≥ L κ ( d ) = L η ( d )+ d we can get(7.71) 1 + c , L η ( d ) h − ≤ c , L − d ≤ θ − / once L > (cid:0) c , log 4 (cid:1) /d .For the second bound, using Lemma 6.9, the first inequality of (4.40) andLemma 5.1(iv),(7.72) k B K k k +1 , ≤ X B ∈B k ( B ′ ) (cid:13)(cid:13) Π X X ∈S k,X ⊃ B | X | k ( R k +1 K )( X ) (cid:13)(cid:13) k +1 , ≤≤ X B ∈B k ( B ′ ) C X X ∈S k,X ⊃ B | X | k k ( R k +1 K )( X ) k k : k +1 ,X,r ≤ X B ∈B k ( B ′ ) X X ∈S k,X ⊃ B C | X | k | X | k k K ( X ) k k,X,r ≤≤ X B ∈B k ( B ′ ) X X ∈S k,X ⊃ B C | X | k | X | k k K k k ( A ) k ≤ L d M k K k k ( A ) k , for any B ′ ∈ B k +1 . Here the factor L d comes from the number of blocks B ∈ B k ( B ′ )and we included into M = M ( d ) the constant C = C ( d ) as well as the bound onthe number of short polymers containing a fixed block. (cid:3) Lemma 7.1 in conjunction with the estimates above give the estimates (4.84) inProposition 4.7. Proof of Remark 4.8. The smoothness of the operators with respect to the fine tuning parameter q follows for B ( q ) and C ( q ) with the corresponding bounds in Chapter 6 and for A ( q ) from the regularity of the finite range decomposition (4.3), i.e., (4.85) follows with C = C ( d, h, L, ω ) > r ≥ ℓ + 3 and all k q k ≤ . (cid:3) HAPTER 8 Fine Tuning of the Initial Conditions Finally, we address the fine tuning Theorem 4.9. First, in Section 8.1, we provethe smoothness of the map F assigning a fixed point of the renormalisation map T to initial values H and K . Then we can specify the map H that chooses theinitial ideal Hamiltonian H in a self-consistent way so that it is reproduced in thefirst component H of F . Its properties summarized in Theorem 4.9 are proven inSection 8.2. F Considering the space E with the norm k·k ζ with ζ > Y r introduced in (4.97) and (4.98), we find a map F from aneighbourhood of origin in E × M (with a shorthand M = M ( B , X )) to Y r sothat T ( F ( K , H ) , K , H ) = F ( K , H ) with the following smoothness properties. Proposition . Let d = 2 , , ω ≥ d d +1 + 1) , r ≥ , and m + 2 ≤ r be fixed and let L , h ( L ) , A ( L ) , M > (see (4.98) ), and θ ∈ (1 / , / be theconstants from Propositions 4.6 and 4.7. Then there exist constants α = α ( M, θ ) ≥ and η = η ( θ ) ∈ (0 , determining the norm of the spaces Y r , r = r , r − , . . . , r − m and, for any L ≥ L , h ≥ h ( L ) , and A ≥ A ( L ) , a constant ζ = ζ ( h ) determining the norm k·k ζ on E and constants b ρ, b ρ , b ρ > so thatthere exists a unique function F : B E × M ( b ρ , b ρ ) → B Y r ( b ρ ) solving the equation T ( F ( K , H ) , K , H ) = F ( K , H ) (see (4.104) ). Moreover, (8.1) F ∈ e C m ( B E × M ( b ρ , b ρ ) , Y ) with bounds on derivatives that are uniform in N , i.e., there is b C such that (8.2) k D j K D ℓ H F ( K , H )( ˙ K , . . . , ˙ K , ˙ H , . . . , ˙ H ) k Y r − ℓ ≤ b C k ˙ Hk ℓ k ˙ Kk jζ , for all ( K , H ) ∈ B E × M ( b ρ , b ρ ) and all ℓ, j ∈ N with ℓ + j ≤ n ≤ m . The proof of Proposition 8.1 is based on Theorem E.1 applied in conjunctionwith Propositions 4.6 and 4.7. Here, the map T : Y × E × M → Y plays therole of the map F and the sequence of spaces Y = Y r ֒ → Y r − ֒ → . . . ֒ → Y r − m ,2 m < r , the role of the sequence X n , n = m, m − , . . . , 0. Using O ρ := B Y ( ρ ), W ρ := B E ( ρ ) = {K ∈ E : kKk ζ ≤ ρ } , and V ρ := {H ∈ M : kHk ≤ ρ } , wejust have to verify the assumptions of Theorem E.1, that is we need to prove thefollowing claim. Lemma . Let L, h , and A be constants as in Proposition 8.1 and let θ ∈ (1 / , / and M > be the constants from Proposition 4.7. Then there existparameters α and η of the norms in Y r depending only on θ and M , constants ρ > , and ζ depending on h and A , so that: 890 8. FINE TUNING OF THE INITIAL CONDITIONS (i) T ∈ e C m ( O ρ × W ρ × V ρ , Y ) with the bounds on corresponding derivatives thatare uniform in N ,(ii) T (0 , , H ) = 0 for all H ∈ V ρ , and(iii) (cid:13)(cid:13)(cid:13) D T ( y , , H ) (cid:12)(cid:12) y =0 (cid:13)(cid:13)(cid:13) L ( Y r , Y r ) ≤ θ for all H ∈ V ρ and r = r , r − , . . . , r − m . Proof. Let us recall the definition of the map T . The 2 N coordinates of theimage(8.3) T ( y , K , H ) = y = ( H , H , K , . . . , H N − , K N − , K N )are defined by(8.4) H k = (cid:0) A ( H ) k (cid:1) − (cid:0) H k +1 − B ( H ) k K k (cid:1) and K k +1 = S k ( H k , K k , H ) , where we set H N = 0 and(8.5) K ( X, ϕ ) := exp n − X x ∈ X H ( x, ϕ ) o Y x ∈ X K ( ∇ ϕ ( x ))with K ∈ E . Notice that A H k , B H k , and S k ( H k , K k , H ) depend on H only throughthe coefficient of its quadratic term q = q ( H ). We will also use a shorthand(8.6) K ( X, ϕ ) =: K ( K , H ) ( X, ϕ ) = Y x ∈ X K ( K , H )0 ( x, ϕ )with(8.7) K ( K , H )0 ( x, ϕ ) = exp (cid:8) −H ( x, ϕ ) (cid:9) K ( ∇ ϕ ( x )) . Here we explicitly invoke the dependence of the map S k on k in contradistinctionto Chapter 6, where the index k was omitted. Notice that the only two coordinatesof y that depend on K (through K ) are H = (cid:0) A ( H )0 (cid:1) − (cid:0) H − B ( H )0 K (cid:1) and K = S ( H , K , H ).(i) The fact that T ∈ e C m ( O ρ × W ρ × V ρ , Y ) follows from Propositions 4.6 and 4.7.We will treat separately the coordinates K k +1 , k = 1 , , . . . , N − 1, the coordinates H k , k = 1 , , . . . , N − 1, and finally, the coordinates H and K that depend on K .Reinstating the dependence on k , we denote more explicitly the sequence ofnormed spaces M k,r = { M ( P c k , X ) : k·k ( A ) k,r < ∞} , r = r , r − , . . . , r − m , aswell as M k, = ( M ( B k , X ) , k·k k, ). Then the claim of Proposition 4.6 is that themaping S k : U k,ρ × V / → M k +1 = M k +1 ,r belongs to e C m ( U k,ρ × V / , M k +1 )for all k = 1 , , . . . , N − 1. Here, U k,ρ = { ( H, K ) ∈ M k, × M k,r : k H k k, < ρ, k K k ( A ) k,r < ρ } For the coordinates H k , k = 1 , , . . . , N − 1, we first observe that the definingmap H k = ( A ( H ) k ) − (cid:0) H k +1 − B ( H ) k K k (cid:1) is linear in H k +1 and K k and that it doesnot depend on K . Consider thus the map(8.8) G : ( y , H ) ( A ( H ) k ) − (cid:0) H k +1 − B ( H ) k K k (cid:1) and verify that G ∈ e C m ( Y × V ρ , M k, ). .1. PROPERTIES OF THE MAP F First, we will address the smoothness of the term B ( H ) k K k . Comparing theformula (4.82) with (6.18), we see that(8.9) B ( H ) k K k ( B ′ , ϕ ) = − X B ∈B ( B ′ ) R (0 , K k , q ( H )) , obtaining the needed smoothness relying on the fact that R ∈ e C m ( U k,ρ ×V ρ , M k, )(see Lemma 6.7) and the fact that the projection H 7→ q ( H ) is a linear mapping.Denoting H = H k +1 − B ( H ) k K k ∈ M k +1 , and rewriting it in terms of the coor-dinates λ, a, c , d we see that the linear operator ( A ( H ) k ) − only shifts the coordinate λ by(8.10) − X x ∈ B d X i,j =1 d i,j ∇ i ∇ ∗ j C ( q ( H )) k +1 (0) , keeping the other coordinates unchanged (cf. the proof of Lemma 7.8). The deriva-tives of this shift can be estimated by finite range decomposition bound (4.3) yield-ing(8.11) sup kHk ≤ (cid:12)(cid:12) ( D ℓ ∇ i ∇ ∗ j C ( q ( H )) k +1 )(0)( ˙ H , . . . , ˙ H ) (cid:12)(cid:12) ≤ c ,ℓ L − kd L η (2 ,d ) k ˙ Hk ℓ where we used that(8.12) 12 d X i,j =1 | d i,j | ≤ h k H k k +1 , according to (4.44). Hence(8.13) k D ℓ (( A ( q ( H )) k ) − H )( ˙ H , . . . , ˙ H ) k k, = k D ℓ G ( y , H )( ˙ H , . . . , ˙ H ) k k, ≤ c ,ℓ L η (2 ,d ) h − k H k k +1 , k ˙ Hk ℓ , for kHk ≤ and y ∈ Y . Actually, in [ AKM13 ] it is shown that ∇ i ∇ ∗ j C ( q ) k +1 (0) isanalytic in q .Finally, we consider the coordinates H and K . Their derivatives with respectto K have to be evaluated by composing the derivatives of H and K with respectto K with the derivatives of K with respect to K . We first deal with the coordinate K which can be viewed as a composition of maps(8.14) F : M × E × M → M , × M ,r and S : ( M × M ,r ) × M → M ,r . Indeed, with(8.15) F ( H , K , H ) = ( H , K ( K , H )0 )we get(8.16) K = S ⋄ F, i.e., K ( H , K , H ) = S ( F ( H , K , H ) , H ) . Here, K ( K , H )0 is the polymer defined in (8.6), where we explicitly denoted the de-pendence on K and H . Now, we apply the Chain Rule according to Theorem D.29 jointly with Re-mark D.30 providing bounds on derivatives that are uniform in N . The needed con-dition S ∈ e C m ( U ,ρ × V / , M ) is just the corresponding claim (4.78) from Propo-sition 4.6. For the map F , there is no grading on the domain space M × E × M ,and we will actually show that F ∈ C m ∗ ( U ,ρ × W ρ × V ρ , M × M ,r ). Indeed,choosing a suitable parameter ζ and ρ , both depending on h , we will prove that thederivative D j D ℓ K ( K , H )0 ( ˙ K j , ˙ H ℓ ) exists and(8.17) (cid:13)(cid:13)(cid:13) D j D ℓ K ( K , H )0 ( ˙ K j , ˙ H ℓ ) (cid:13)(cid:13)(cid:13) ,r ≤ C k ˙ Kk jζ k ˙ Hk ℓ for any j, ℓ ≤ m + 1 with C = C ( h, A , m ), and thus also(8.18) lim ( K ′ , H ′ ) → ( K , H ) (cid:13)(cid:13)(cid:13) D j D ℓ K ( K , H )0 ( ˙ K j , ˙ H ℓ ) − D j D ℓ K ( K , H )0 ( ˙ K j , ˙ H ℓ ) (cid:13)(cid:13)(cid:13) ,r = 0for any j, ℓ ≤ m and any ( H , K , H ) ∈ U ,ρ × W ρ × V ρ .Indeed, in view of the product form in (8.5) and (8.6), we first have(8.19) D ℓ K ( X, ϕ )( ˙ H , . . . , ˙ H ) = X k ∈ N X P x ∈ X kx = ℓ ( − ℓ ℓ ! Q x ∈ X k x ! Y x ∈ X (cid:16) ˙ H ( x, ϕ ) k x e −H ( x,ϕ ) K ( ∇ ϕ ( x )) (cid:17) , and thus(8.20) D j D ℓ K ( K , H )0 ( ˙ K j , ˙ H ℓ ) = X k ∈ N X P x ∈ X kx = ℓ X Y ⊂ X | Y | = j ( − ℓ ℓ ! Q x ∈ X k x ! Y x ∈ X (cid:16) ˙ H ( x, ϕ ) k x e −H ( x,ϕ ) (cid:17) × Y y ∈ Y ˙ K ( ∇ ϕ ( y )) Y y ∈ X \ Y K ( ∇ ϕ ( y ))= X k ∈ N X P x ∈ X kx = ℓ X Y ⊂ X | Y | = j ( − ℓ ℓ ! Q x ∈ X k x ! Y x ∈ X (cid:16) ˙ H ( x, ϕ ) k x (cid:17) Y x ∈ Y ˙ K ( K , H )0 ( x, ϕ ) Y x ∈ X \ Y K ( K , H )0 ( x, ϕ ) . Here, we use the shorthand ˙ K ( K , H )0 ( x, ϕ ) = exp (cid:8) −H ( x, ϕ ) (cid:9) ˙ K ( ∇ ϕ ( x )). Observ-ing that, in the case k = 0, the unit blocks are actually single sites, B k (Λ N ) = Λ N ,we can apply the claim (iia) of Lemma 5.1 to get(8.21) (cid:13)(cid:13)(cid:13) Y y ∈ Y ˙ K ( H , ˙ H ,k y )0 ( y, ϕ ) Y y ∈ X \ Y K ( H , ˙ H ,k y )0 ( y, ϕ ) (cid:13)(cid:13)(cid:13) ,X,r ≤ Y y ∈ Y |k ˙ K ( H , ˙ H ,k y )0 k| , { y } Y y ∈ X \ Y |kK ( H , ˙ H ,k y )0 k| , { y } . Here we introduced the shorthands(8.22) K ( H , ˙ H ,k y )0 ( y, ϕ ) = − ˙ H ( y, ϕ ) k y K ( K , H )0 ( y, ϕ )and(8.23) ˙ K ( H , ˙ H ,k y )0 ( y, ϕ ) = − ˙ H ( y, ϕ ) k y ˙ K ( K , H )0 ( y, ϕ ) . Further, using definitions (4.30) and (4.27),(8.24) |kK ( H , ˙ H ,k y y )0 k| , { y } = sup ϕ | K ( H , ˙ H ,k y )0 ( y, ϕ ) | , { y } ,r exp {− G ,y ( ϕ ) } .1. PROPERTIES OF THE MAP F with the weight function G ,y ( ϕ ) defined in (4.29) and(8.25) | K ( H , ˙ H ,k y )0 ( y, ϕ ) | , { y } ,r = r X r =0 r ! sup | ˙ ϕ | , { y } ≤ (cid:12)(cid:12) D r K ( H , ˙ H ,k y )0 ( y, ϕ )( ˙ ϕ, . . . , ˙ ϕ ) (cid:12)(cid:12) . Using the definition (4.21), we can bound(8.26) | ˙ ϕ | , { y } = max ≤ s ≤ sup w ∈{ y } ∗ h (cid:12)(cid:12) ∇ s ˙ ϕ ( w ) (cid:12)(cid:12) ≥ max (cid:0) h |∇ ˙ ϕ ( y ) | , h |∇ ˙ ϕ ( y ) | (cid:1) . Now(8.27)sup | ˙ ϕ | , { y } ≤ (cid:12)(cid:12) D r K ( H , ˙ H ,k y )0 ( y, ϕ )( ˙ ϕ, . . . , ˙ ϕ ) (cid:12)(cid:12) ≤ sup | ˙ ϕ | , { y } ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d r K ( H , ˙ H ,k y )0 ( y, ϕ + t ˙ ϕ )d t r (cid:12)(cid:12)(cid:12) t =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Defining v = ∇ ϕ ( y ) , w = ∇ ϕ ( y ), and z = h (cid:0) | v | + | w | (cid:1) / r K ( H , ˙ H ,k y )0 ( y, ϕ + t ˙ ϕ )d t r is a sum of terms of the form(8.28)( ˙ λ + ˙ av + h ˙ q v, v i + ˙ cw ) i ( ˙ a ˙ v + h ˙ q v, ˙ v i + ˙ c ˙ w ) i h ˙ q ˙ v, ˙ v i i ( a ˙ v + h q v, ˙ v i + z ˙ w ) j h q ˙ v, ˙ v i j ×× exp {− ( λ + av + h q v, v i + cw ) } d s K ( v + t ˙ v )d t s (cid:12)(cid:12)(cid:12) t =0 such that i + i + i = k y and i + 2 i + j + 2 j + s = r . Using the definition ofthe norm kHk and the fact that h max( | ˙ v | , | ˙ w | ) ≤ | ˙ ϕ | , { y } ≤ 1, the absolute valueof the prefactor above can be bounded by2 i + i + j + j k ˙ Hk j + j (cid:0) z (cid:1) i + i + j Now assume that(8.29) kHk ≤ e ρ ≤ . Since k y ≤ m + 1 and j ≤ m + 1 we have(8.30) (1 + z ) i + i + j ≤ (1 + z ) m +1) ≤ (cid:0) m + 1) e ρ (cid:1) m +1) exp { e ρz } . In the last inequality we used that for a > z ≥ z ) a ≤ (cid:0) a e ρ (cid:1) a/ exp { e ρz } To see this observe that for a > t (1 + t ) a exp {− e ρt } for t ≥ t = t = 12 (cid:0)r a e ρ − (cid:1) and is bounded by (1 + t ) a ≤ (1 + 2 t ) a = (cid:0) a e ρ (cid:1) a/ . As a result, there exists a constant C ( r ) so that for | ˙ ϕ | ≤ | ˙ v | ≤ h ,we have(8.33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d r K ( H , ˙ H ,k y )0 ( ϕ + t ˙ ϕ )d t r (cid:12)(cid:12)(cid:12) t =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( r ) (cid:0) m +1) e ρ (cid:1) m +1) k ˙ Hk k y ×× exp { e ρ | z | } (cid:18) r X s =0 (cid:12)(cid:12)(cid:12)(cid:12) d s K ( v + t ˙ v )d t s (cid:12)(cid:12)(cid:12) t =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ C ( r ) (cid:0) m +1) e ρ (cid:1) m +1) k ˙ Hk k y exp { e ρz } X | α |≤ r h | α | | ∂ αv K ( v ) | for any kHk ≤ e ρ , and any r ≤ r . Finally, choosing(8.34) ζ ≥ h and taking into account that(8.35) G ,y ( ϕ ) ≥ h |∇ ϕ ( y ) | + 1 h |∇ ϕ ( y ) | = z and the definition (2.21) of the norm kKk ζ and using | v | ≤ hz we get(8.36) |kK ( H , ˙ H ,ky ) k| , { y } ≤ e C k ˙ Hk k y sup z ≥ (cid:16) exp { ( e ρ − z } exp { ζ − h z }kKk ζ (cid:17) with(8.37) e C = e C ( r , m, h, e ρ ) = C ( r ) (cid:0) m + 1) e ρ (cid:1) m +1) . The same estimate holds for ˙ K ( H , ˙ H ,ky ) if we replace kKk ζ on the right hand sideby k ˙ Kk ζ . The exponential term can be controlled if for given h we choose ζ and e ρ such that(8.38) h ζ + e ρ ≤ . In particular we may take(8.39) e ρ = 12 and ζ = √ h. Note that (8.38) implies (8.34) and (8.29).Summarising, we get,(8.40) (cid:13)(cid:13)(cid:13) Y y ∈ Y ˙ K ( H , ˙ H ,k y )0 ( ∇ ϕ ( y )) Y y ∈ X \ Y K ( H , ˙ H ,k y )0 ( y, ϕ ) (cid:13)(cid:13)(cid:13) ,X,r ≤≤ e C | X | k ˙ Hk ℓ k ˙ Kk jζ kKk | X |− jζ . Since ℓ ≤ m + 1 the sum in (8.20) over k ∈ N X with P x ∈ X k x = ℓ involves atmost ( m + 2) | X | terms. The sum over Y involves at most 2 | X | terms. The countingterms with the factorial in (8.20) are bounded by ( m + 1)!. Thus (8.20) and (8.40)give(8.41) k D j D ℓ K ( X, K , H , ˙ K , . . . , ˙ K , ˙ H , . . . , ˙ H ) k ,r ≤ ( m + 1)!(2( m + 2)) | X | e C | X | kKk X − jζ k ˙ Kk jζ k ˙ Hk ℓ . .1. PROPERTIES OF THE MAP F Thus with ζ = √ h we have for all K ∈ B E ( ρ ) with(8.42) ρ = ρ ( A ) = (cid:0) m + 2)( m + 1)! A e C (cid:1) − and all H ∈ B M ( e ρ ) with e ρ = ,(8.43) Γ A ( X ) k D j D ℓ K ( X, K , H , ˙ K , . . . , ˙ K , ˙ H , . . . , ˙ H ) k ,r ≤ C k ˙ Kk jζ kHk ℓ with(8.44) C = C ( A , m ) = (cid:0) ( m + 1)!(2( m + 2) e C A (cid:1) m +1 . Finally, for the coordinate H = ( A ( H ) ) − (cid:0) H − B ( H )0 K (cid:1) , we can again applythe Chain Rule according to Theorem D.29. The image coordinate H is obtainedas a composition of maps(8.45) F : M , × E × M → M , × M ,r and G : ( M , × M ,r ) × M → M ,r with(8.46) F ( H , K , H ) = ( H , K ( K , H )0 ) and G (( H , K ) , H ) = ( A ( H )0 ) − (cid:0) H − B ( H )0 K (cid:1) yielding H = G ⋄ F . Both needed conditions, G ∈ e C m ( Y × V ρ , M ,r ) as well as F ∈ C m ∗ ( U ,ρ × W ρ × V ρ , M , × M ,r ) have been already proven.(ii) This is an immediate consequence of the definition of the map T and the factthat S (0 , , H ) = 0 (cf. (4.62)).(iii) Using that K = 0 for K = 0 and that ∂S k ∂H k (0 , , H ) = ∂S k ∂K k (0 , , H ) = 0, we cancompute the derivatives of y = T ( y , , H ) at H = 0: ∂H k ∂H j = ( A − k if j = k + 1 , j = 0 , . . . , N − 20 otherwise, ∂H k ∂K j = ( − A − k B k if j = k, , (8.47)and ∂K k +1 ∂H j = 0 ,∂K k +1 ∂K j = ( C k if j = k = 0 , k, j = 0 , . . . , N − y ∈ Y r with k y k Y r ≤ y under the map ∂ T ( y , , H ) ∂ y (cid:12)(cid:12) y =0 ,(8.49) y = ∂ T ( y , , H ) ∂ y (cid:12)(cid:12)(cid:12) y =0 y . Since k y k Y r ≤ 1, we have k H ( y ) k k k, ≤ η k , k = 0 , . . . , N − 1, and k K ( y ) k k k,r ≤ η k α , k = 1 , . . . , N , for the coordinates H ( y ) k , K ( y ) k of the vector y . Using H ( y ) k , K ( y ) k , forthe coordinates of the image y , we get k H ( y )0 k k, ≤ k A − k η ; k H ( y ) k k k, ≤ k A − k k η k +1 + k A − k kk B k k η k α ≤ η k √ θ ( η + Mα ) , k = 1 , . . . , N − k H ( y ) N − k N − , ≤ k A − N − kk B N − k η N − α ≤ η N − Mα √ θ ; k K ( y )1 k k,r = 0; k K ( y ) k k k,r ≤ k C k − k η k α ≤ θ η k α , k = 2 , . . . , N. As a result, k y k Y r ≤ (cid:0) √ θ ( η + Mα ) (cid:1) ∨ θη . It suffices to choose the parameters η and α so that η + M/α ≤ θ / ( θ < η < θ / ),yielding(8.50) (cid:13)(cid:13)(cid:13) ∂ T ( y , , H ) ∂ y (cid:12)(cid:12)(cid:12) y =0 (cid:13)(cid:13)(cid:13) L ( Z s , Z s ) ≤ θ < , s = r , r − , . . . , r − . (cid:3) Proof of Proposition 8.1. Having thus, in Lemma 8.2, verified the assump-tions (E.1)-(E.4) of Theorem E.1 for the map T in the role of F , there exist con-stants b ρ , b ρ , and b ρ depending (through ρ in Lemma 8.2) on h and A and b C ,depending (through C = C ( L, h, A ) in Proposition 4.6) on L, h, and A , and themap(8.51) F : B E × M ( b ρ , b ρ ) → B Y r ( b ρ )(in the role of f ) so that T ( F ( K , H ) , K , H ) = F ( K , H ) for any( K , H ) ∈ B E × M ( b ρ , b ρ ) , and(8.52) F ∈ e C m ( B E × M ( b ρ , b ρ ) , Y ) , satisfying (8.2) whenever ( K , H ) ∈ B E × M ( b ρ , b ρ ) and j, ℓ ∈ N with ℓ + j ≤ m .Here, the estimates (8.2) follow from the bounds (E.8). (cid:3) H Using our results in the previous section we finally obtain a map H mappinga neighbourhood of the origin in E to M so that T ( F ( K , H ( K )) , K , H ( K )) = F ( K , H ( K )) and Π( F ( K , H ( K ))) = H ( K ). This requires another application ofthe implicit function theorem, this time for the composition of the projection Πwith the map F in Proposition 8.1. We write G := Π ◦ F in the following. Theprojection Π : Y r − n → M is a bounded linear mapping for any 0 ≤ n ≤ m .Using Proposition 8.1 we obtain, in particular, that G ∈ C m ∗ ( B E × M ( b ρ , b ρ ) , M ).Note that F (0 , H ) = 0 because T (0 , , H ) = 0 for all H ∈ V ρ (see (ii) in Lemma 8.2),and thus G (0 , H ) = 0 and D H G (0 , 0) = 0. Therefore, by standard implicit function .2. PROPERTIES OF THE MAP H theorem, there exists a C m ∗ -map H : B E ( ρ ) → B M ( ρ ) with a suitable ρ ≤ ˆ ρ and ρ = ˆ ρ such that G ( K , H ( K )) = H ( K ).PPENDIX A Discrete Sobolev Estimates For the convenience of the reader we recall a discrete version of the Sobolevinequality. Discrete Sobolev inequalities are classical, see, e.g., Sobolev’s originalwork [ Sob40 ]. Let B n = [0 , n ] d ∩ Z d , and for p > k f k p = k f k p,B n = (cid:16) X x ∈ B n | f ( x ) | p (cid:17) /p for any function f : B n → R . Proposition A.1 . For every p ≥ and m, M ∈ N there exists a constant C = C ( p, M, m ) such that:(i) If ≤ p ≤ d , p ∗ = p − d , and q ≤ p ∗ , q < ∞ , then (A.2) n − dq k f k q ≤ C n − d k f k + C n − dp k∇ f k p . (ii) If p > d , then (A.3) (cid:12)(cid:12) f ( x ) − f ( y ) (cid:12)(cid:12) ≤ C n − dp k∇ f k p for all x, y ∈ B n . (iii) If m ∈ N , ≤ p ≤ dm , p m = p − md , and q ≤ p m , q < ∞ , then (A.4) n − dq k f k q ≤ C n − d M − X k =0 k ( n ∇ ) k f k + C n − dp k ( n ∇ ) M f k p . (iv) If M = ⌊ d +22 ⌋ , the integer value of d +22 , then (A.5) max x ∈ B n | f ( x ) | ≤ C n − d M X k =0 k ( n ∇ ) k f k . Remark A.2 . (i) In the proof of (iv) we actually get(A.6) max x ∈ B n | f ( x ) | ≤ ( n + 1) − d X x ∈ B n | f ( x ) | + C n − d M X k =1 k ( n ∇ ) k f k . (ii) As written, the higher derivatives on the RHS of (i)-(iv) require the values of f outside B n . If one traces the dependence more carefully then one sees that( ∇ α . . . ∇ α d d f )( x ) is only needed for x such that x + α e + · · · + α d e d ∈ B n , sothat only the values of f inside B n are needed. The proof may be reduced to the continuous case by interpolation. Let n = 1, B = { , } d , f : B → R + , and let e f be the interpolation of f which is affine ineach coordinate direction, i.e., e f is the unique function of the form(A.7) e f ( x ) = d Y i =1 ( a i x i + b i ) , e f ( x ) = f ( x ) for x ∈ { , } d . The Proposition A.1 will be proven with help of the following Lemma. Lemma A.3 . (i) p +1) d d P x ∈ B f p ( x ) ≤ R (0 , d e f p ( x )d x ≤ d P x ∈ B f p ( x ) . (ii) sup x ∈ (0 , d | ∂ i e f ( x ) | ≤ max x ∈ B ,x i =0 | f ( x + e i ) − f ( x ) | ≤ (cid:16)P x ∈ B ,x i =0 | f ( x + e i ) − f ( x ) | p (cid:17) /p for any i = 1 , . . . , d . Proof. (i) The integrand is a product of functions of one variable. Takinginto account that(A.8) 12 d X x ∈ B f p ( x ) = d Y i =1 (cid:16) 12 ( a i + b i ) p + 12 b pi (cid:17) , it suffices to prove the claim for d = 1. Considering thus a nonnegative function onthe interval [0 , 1] of the form ax + b and assuming w.l.o.g. that a, b ≥ 0, we get(A.9) Z ( ax + b ) p d x = p X k =0 (cid:18) pk (cid:19) k + 1 a k b p − k ≤ b p + p X k =1 (cid:18) pk (cid:19) a k b p − k = 12 b p + 12 ( a + b ) p . On the other hand,(A.10) p X k =0 (cid:18) pk (cid:19) k + 1 a k b p − k ≥ p + 1 p X k =0 (cid:18) pk (cid:19) a k b p − k = 1 p + 1 ( a + b ) p ≥ p + 1 (cid:16) 12 ( a + b ) p + 12 b p (cid:17) . (ii) For e f of the form (A.7) we have ∂ i e f ( x ) = a i Q dj = i ( a j x j + b j ) while, on theother hand, we have a i Q dj = i ( a j x j + b j ) = e f ( x + e i ) − e f ( x ) = f ( x + e i ) − f ( x ) forany x ∈ B such that x i = 0. (cid:3) Proof of Proposition A.1. (i) and (ii) follow from Lemma A.3 and thecontinuous embedding Theorem.The claim (iii) follows from (i) by iteration.To prove (iv), assume first that d is odd and thus M = ⌊ d +22 ⌋ = d + . Let usapply (iii) with p = 2, m = M − 1, and(A.11) 1 p m = 12 − M − d = d − ( d − d = 12 d . Hence,(A.12) n − d d k∇ f k d ≤ C n − d − M X k =1 k ( n ∇ ) k f k . . DISCRETE SOBOLEV ESTIMATES 101 Further,(A.13) (cid:12)(cid:12) f ( x ) − f ( y ) (cid:12)(cid:12) ≤ C n − d d k∇ f k d = C n k∇ f k d for all x, y ∈ B n by (ii). Averaging over y yields(A.14) (cid:12)(cid:12) f ( x ) − ( n + 1) − d X y ∈ B n f ( y ) (cid:12)(cid:12) ≤ C n k∇ f k d . On the other hand,(A.15) (cid:12)(cid:12) ( n + 1) − d X y ∈ B n f ( y ) (cid:12)(cid:12) ≤ ( n + 1) − d (cid:16) X y ∈ B n f ( y ) (cid:17) / (cid:16) X y ∈ B n (cid:17) / ≤ ( n + 1) − d/ k f k yielding(A.16) | f ( x ) | ≤ C n k∇ f k d + ( n + 1) − d/ k f k for all x ∈ B n . The assertion (iv) for odd d follows.Similarly for even d when M = ⌊ d +22 ⌋ = d + 1 and we use m = M − q = 2 d > p m = d . (cid:3) PPENDIX B Integration by Parts and Estimates of theBoundary Terms For the convenience on the reader we spell out the estimates of the boundaryterms in detail. a) d = 1The forward and backward derivative are ∂v ( x ) = v ( x + 1) − v ( x ) and ∂ ∗ v ( x ) = v ( x − − v ( x ). Proposition B.1 ( Integration by parts ) . Let g, v, u : Z → R and m ∈ N .Then:(i) m X x = − m g ( x ) ∂v ( x ) = m X x = − m ∂ ∗ g ( x ) v ( x ) + g ( m ) v ( m + 1) − g ( − m − v ( − m ) . (ii) m X x = − m ∂u ( x ) ∂v ( x ) = m X x = − m ( ∂ ∗ ∂u )( x ) v ( x ) + ∂u ( m ) v ( m + 1) − ∂u ( − m − v ( − m ) . Proposition B.2 ( Evaluation of the boundary terms ) . There exist aconstant c < √ such that for any v : Z → R and any m ∈ N , m > , one has (B.1) v ( − m ) + v ( m + 1) ≤ c m + 1 m X x = − m v ( x ) + c (2 m + 1) m X x = − m ∂v ( x ) . Proof. Assume first that the number of those x ∈ {− m, . . . , m } for which v ( x ) ≥ (cid:0) v ( − m ) + v ( m + 1) (cid:1) is at least m +1 √ . Then P mx = − m v ( x ) ≥ √ (2 m +1) (cid:0) v ( − m ) + v ( m + 1) (cid:1) .On the other hand, if the number of such x ’s is less then m +1 √ , then thereexists x such that ∂v ( x ) ≥ √ v ( − m ) + v ( m +1) m +1 , implying m X x = − m ∂v ( x ) ≥ √ v ( − m ) + v ( m + 1) m + 1 . Indeed, having assured the existence of y and z such v ( y ) < (cid:0) v ( − m ) + v ( m +1) (cid:1) (the existence of such y is obvious for m > (cid:0) − √ (cid:1) (2 m +1) > 1) and v ( z ) ≥ (cid:0) v ( − m ) + v ( m + 1) (cid:1) (again, its existence follows since (cid:0) v ( − m ) + v ( m + 1) (cid:1) ≤ max (cid:8) v ( − m ) , v ( m + 1) (cid:9) ) implying that the interval (cid:2) (cid:0) v ( − m ) + v ( m + 1) (cid:1) , (cid:0) v ( − m ) + v ( m + 1) (cid:1)(cid:3) has to be spanned within at most m +1 √ increments ∂v ( x ) .In both cases,(B.2) 12 m + 1 m X x = − m v ( x ) + (2 m + 1) m X x = − m ∂v ( x ) ≥ √ (cid:0) v ( − m ) + v ( m + 1) (cid:1) implying the claim. (cid:3) The combination of Proposition B.1 and B.2 yields: Proposition B.3 . Let u, v : Z → R and m ∈ N . With the constant c fromProposition B.2 and any η > , one has (B.3) (cid:12)(cid:12)(cid:12) m X x = − m ∂u ( x ) ∂v ( x ) (cid:12)(cid:12)(cid:12) ≤ 12 (2 m +1) η m X x = − m (cid:12)(cid:12) ( ∂ ∗ ∂u )( x ) (cid:12)(cid:12) + 12 η (2 m + 1) m X x = − m v ( x ) ++ 2 m + 12 η h ∂u ( − m − + ∂u ( m ) i + c η h m + 1) m X x = − m v ( x ) + m X x = − m ∂v ( x ) i . b) Multidimensional case Let X ∈ P k be a union of k -blocks. Further, let ∂ ± X = ∪ di =1 ∂ ± i X , where, forany i = 1 , . . . , d ,(B.4) ∂ − i X := { x ∈ Z d : x / ∈ X, x + e i ∈ X or x ∈ X, x + e i / ∈ X } and(B.5) ∂ + i X = ∂ − i X + e i := { x + e i : x ∈ ∂ − i X } . Notice that ∂ − X ∪ ∂ + X = ∂X , the boundary defined in (4.31). Lemma B.4 . Let B be a k -block and let v : B ∪ ∂B → R . Then, for any i = 1 , . . . , d , (B.6) X x ∈ ∂ + i B v ( x ) ≤ c (cid:16) L k X x ∈ B v ( x ) + L k X x ∈ B |∇ i v ( x ) | (cid:17) and (B.7) X x ∈ ∂ − i B v ( x ) ≤ c (cid:16) L k X x ∈ B v ( x ) + L k X x ∈ B |∇ ∗ i v ( x ) | (cid:17) , where c is the constant from Proposition B.2. Proof. Applying Proposition B.2 to all lines in B that are parallel to e i , weget (B.6). Similarly for (B.7), when considering the sites on these lines in theopposite order. (cid:3) Notice that, using ∇ ∗ i v ( x ) = −∇ i v ( x − e i ), the last term in (B.7) can be actuallyreplaced by L k P x ∈ B − e i |∇ i v ( x ) | To formulate the following immediate corollary of Lemma B.4, let, for any X ∈ P k and ℓ ∈ N , the neighbourhood U ℓ ( X ) be defined iteratively with U ( X ) = X ∪ ∂X and U ℓ +1 ( X ) = U ℓ ( X ) ∪ ∂U ℓ ( X ). ) MULTIDIMENSIONAL CASE 105 Proposition B.5 . Let X ∈ P k and u : U ( X ) → R . With the constant c fromProposition B.2,(a) L k X x ∈ ∂X |∇ v ( x ) | ≤ c (cid:16) X x ∈ X |∇ v ( x ) | + L k X x ∈ U ( X ) |∇ v ( x ) | (cid:17) , (b) L k X x ∈ ∂X |∇ v ( x ) | ≤ c (cid:16) L k X x ∈ X |∇ v ( x ) | + L k X x ∈ U ( X ) |∇ v ( x ) | (cid:17) , and(c) L k X x ∈ ∂X |∇ v ( x ) | ≤ c (cid:16) L k X x ∈ X |∇ v ( x ) | + L k X x ∈ U ( X ) |∇ v ( x ) | (cid:17) . Proof. Let B , . . . , B n denote the k -blocks contained in X . Applying LemmaB.4 to each B ℓ , ℓ = 1 , . . . , n , i = 1 , . . . , d , observing that(B.8) ∂X ⊂ n [ ℓ =1 ∂B ℓ , and summing over i , we get(B.9) L k X x ∈ ∂X |∇ v ( x ) | ≤ c (cid:16) X x ∈ X |∇ v ( x ) | + L k X x ∈ X d X i =1 (cid:0) |∇ i v ( x ) | + |∇ ∗ i ∇ i v ( x ) | (cid:1)(cid:17) . Using(B.10) X x ∈ X d X i =1 |∇ ∗ i ∇ i v ( x ) | = X x ∈ X − e i d X i =1 |∇ i v ( x ) | ≤ X x ∈ U ( X ) |∇ v ( x ) | , we get the first claim.The second and the third claim follow in a similar way. (cid:3) Notice that the sums over x ∈ U ( X ) on the right hand side of the bounds inProposition B.5 can be actually replaced by the sums over x ∈ ( X ∪ ∂ − X ) \ ( X ∩ ∂ − X ). Proposition B.6 . Let u, v : X ∪ ∂X → R and X ∈ P k . With the constant c from Proposition B.2 and any η > , we get (B.11) (cid:12)(cid:12) X x ∈ X ∇ u ( x ) ∇ v ( x ) (cid:12)(cid:12) ≤ η (1 + c d )2 L k X x ∈ X ∪ ∂ − X v ( x ) + L k η X x ∈ ∂ − X |∇ u ( x ) | ++ c η X x ∈ X |∇ v ( x ) | + L k η X x ∈ X ∪ ∂ − X |∇ u ( x ) | . Proof. For any x ∈ ∂ − i X , let ǫ i ( x ) = +1 if x ∈ X and ǫ i ( x ) = − x X .By Proposition B.1, for each i ∈ { , . . . , d } , we have(B.12) X x ∈ X ∇ i u ( x ) ∇ i v ( x ) = X x ∈ X ∇ ∗ i ∇ i u ( x ) v ( x ) + X x ∈ ∂ − i X ǫ i ( x ) ∇ i u ( x ) v ( x + e i ) . 06 B. INTEGRATION BY PARTS AND ESTIMATES OF THE BOUNDARY TERMS Summing over i = 1 , . . . , d , we get(B.13) (cid:12)(cid:12) X x ∈ X ∇ u ( x ) ∇ v ( x ) (cid:12)(cid:12) ≤ d X i =1 X x ∈ X − e i |∇ i u ( x ) v ( x ) | + d X i =1 X x ∈ ∂ − i X |∇ i u ( x ) v ( x + e i ) | ≤≤ L k η X x ∈ X ∪ ∂ − X |∇ u ( x ) | + η L k d X i =1 X x ∈ X − e i v ( x ) + L k η X x ∈ ∂ − X |∇ u ( x ) | + η L k d X i =1 X x ∈ ∂ + i X v ( x ) . Applying now Lemma B.4 on the last term, we get the claim. (cid:3) Lemma B.7 . Let Y ⊂ X , X, Y ∈ P k , and u : U ( X ) → R . Then (B.14) max x ∈ X u ( x ) ≤ | Y | X x ∈ Y u ( x ) + 2(diam X ) max x ∈ X |∇ u ( x ) | . Proof. Cf. [ Bry09 , Lemma 6.20]. Considering the shortest path from any x ∈ X to y ∈ Y , we have(B.15) | u ( x ) | ≤ | u ( y ) | + | x − y | ∞ max z ∈ X |∇ u ( z ) | . Using that | x − y | ∞ ≤ diam X (with the diameter taken in | · | ∞ metric on Z d ), usingthe inequality ( a + b ) ≤ a + 2 b , and averaging both sides over Y , we get(B.16) u ( x ) ≤ | Y | X y ∈ Y u ( y ) + 2(diam X ) max z ∈ X |∇ u ( z ) | yielding the claim. (cid:3) PPENDIX C Gaussian Calculus Here we recall the formulae for the derivative of a Gaussian integral with respectto the covariance matrix. The arguments are classical, but we provide proofs forthe convenience of the reader. We begin with the first derivative. We will make thefollowing general assumptions throughout this appendix. Let V be a finite dimensional Euclidean vector space with scalar product ( · , · ) and Lebesgue measure λ . Denote by Sym (+) ( V ) and Sym ( ≥ ) ( V ) the set of positivedefinite respectively of positive semi-definite symmetric operators on V . For C ∈ Sym (+) ( V ) denote by µ C the Gaussian measure with covariance C . Let g : V → R bemeasurable and assume that there exists a B ∈ Sym ( ≥ ) ( V ) and a constant M ∈ R such that | g ( x ) | ≤ M e ( B x,x ) for all x ∈ V. For C − > B define(C.1) H ( C ) := Z V g ( x ) µ C (d x ) = 1det(2 π C ) / Z V g ( x )e − ( C − x,x ) λ (d x ) . We first recall that H is real-analytic in the set { C ∈ Sym (+) ( V ) : C − > B } . Infact we will extend H to a complex analytic function as follows. Let e V denote thecomplexification of V with the canonical sesquilinear-form ( · , · ), let GL ( e V ) denotethe set of all invertible C -linear maps from e V to itself and let U := { C ∈ GL ( e V ) : Re ( C − x, x ) > ( B x, x ) ∀ x ∈ V \ { }} . Define H on U by the right hand side of (C.1). Lemma C.1 . (i) The map H : U → C is analytic and the derivative at C in direction ˙ C reads as (C.2) DH ( C , ˙ C ) = Z V g ( x ) 12 (cid:0) ( C − ˙ CC − x, x ) − Tr( C − ˙ C ) (cid:1) µ C (d x ) . (ii) Assume in addition that g is continuous and that there exists a continuousfunction w : V → (0 , ∞ ) such that (C.3) g ( x + y ) ≤ M e ( B x,x ) w ( y ) , x, y ∈ V. Define (C.4) e H ( C )( y ) := Z V g ( x + y ) µ C (d x ) for all y ∈ V. Then e H is an analytic map from U to the space C w := { h ∈ C ( V ) : k h k w < ∞} , where k h k w := sup y ∈ V | h ( y ) || w ( y ) | , and the derivative at C in direction ˙ C ∈ GL ( e V ) is given as D e H ( C , ˙ C )( y ) = Z V g ( x + y ) D f ( C , x, ˙ C ) λ (d x ) , y ∈ V, where f ( C , x ) := e − ( C − x,x ) det(2 π C ) / . Proof. (i) Set(C.5) f ( C , x ) := e − ( C − x,x ) det(2 π C ) / . Then for every x ∈ V the map C f ( C , x ) is complex differentiable in U , and(using Jacobi’s formula for the derivative of determinants) we get that(C.6) D f ( C , x, ˙ C ) = 12 (cid:0) ( C − ˙ CC − x, x ) − Tr( C − ˙ C ) (cid:1) f ( C , x ) . In particular for each ε > M ′ > (cid:12)(cid:12) D f ( C , x, ˙ C ) (cid:12)(cid:12) ≤ M ′ e ε | x | e − ( C − x,x ) | ˙ C | . Since Re ( C − ) > B and since V is finite-dimensional we also have that Re ( C − ) > B + ε Id and thus the function g ( x ) (cid:12)(cid:12) D f ( C , x, ˙ C ) (cid:12)(cid:12) is integrable. Now for any ˙ C = 0 we estimate(C.8) 1 | ˙ C | (cid:12)(cid:12)(cid:12) H ( C + ˙ C ) − H ( C ) − Z V g ( x ) D f ( C , x, ˙ C ) λ (d x ) (cid:12)(cid:12)(cid:12) ≤ Z V | g ( x ) | (cid:12)(cid:12)(cid:12) f ( C + ˙ C ) − f ( C ) − D f ( C , x, ˙ C ) | ˙ C | (cid:12)(cid:12)(cid:12) λ (d x ) . For ˙ C → x ∈ V . It remains to find an integrable majorant. We have f ( C + ˙ C , x ) − f ( C , x ) = Z D f ( C + s ˙ C , x ) d s. Now for every C ∈ U and every ε > δ > M ′′ > e C ∈ B δ ( C ) we have (cid:12)(cid:12) D f ( e C , x, ˙ C ) (cid:12)(cid:12) ≤ M ′′ e ε | x | e − ( C − x,x ) | ˙ C | . Hence for | ˙ C | < δ the integrand in (C.8) is bounded by the integrable function | g ( x ) | ( M ′ + M ′′ )e ε | x | e − ( C − x,x ) . Thus by the dominated convergence theorem the right hand side of (C.8) goes tozero as ˙ C → 0. This concludes the proof of (i). . GAUSSIAN CALCULUS 109 (ii) The continuity of the map y e H ( C )( y ) follows directly from the dominatedconvergence theorem. Indeed, assume that y k → y in V as k → ∞ . Using thecontinuity of g we obtain g ( x + y k ) f ( C , x ) → g ( x + y ) f ( C , x ) for every x ∈ V as k → ∞ . Moreover, for | y k − y | < δ we have (cid:12)(cid:12) g ( x + y k ) f ( C , x ) (cid:12)(cid:12) ≤ M e ( B x,x ) (cid:0) sup z ∈ B δ ( y ) w ( z ) (cid:1) f ( C , x ) , and the right hand side is integrable. Hence e H ( C )( y k ) → e H ( C )( y ) as k → ∞ by the dominated convergence theorem. To verify complex differentiability definefirst the linear map ( L ˙ C )( y ) := Z V g ( x + y ) D f ( C , x, ˙ C ) λ (d x ) . Then one sees as above that y ( L ˙ C )( y ) is continuous. Moreover it follows fromthe bounds (C.3) and (C.7) that k L ˙ C k w ≤ M M ′ | ˙ C | Z V e (( B + ε Id − C − ) x,x ) λ (d x ) < ∞ . Thus L is a bounded linear map from GL ( e V ) to C w ( V ). Finally we check differen-tiability. We have (cid:12)(cid:12)(cid:12) e H ( C + ˙ C )( y ) − e H ( C )( y ) − L ˙ C ( y ) (cid:12)(cid:12)(cid:12) ≤ Z V | g ( x + y ) | (cid:12)(cid:12) f ( C + ˙ C , x ) − f ( C , x ) − D f ( C , x, ˙ C ) (cid:12)(cid:12) λ (d x ) ≤ M w ( y ) Z V e ( B x,x ) (cid:12)(cid:12) f ( C + ˙ C , x ) − f ( C , x ) − D f ( C , x, ˙ C ) (cid:12)(cid:12) λ (d x ) . Dividing by w ( y ) | ˙ C | and taking the supremum over y we get k e H ( C + ˙ C ) + e H ( C ) − L ˙ C k w ≤ M Z V e ( B x,x ) (cid:12)(cid:12) f ( C + ˙ C , x ) − f ( C , x ) − D f ( C , x, ˙ C ) (cid:12)(cid:12) | ˙ C | λ (d x ) . Now as in (i) it follows from the dominated convergence theorem that the right handside goes to zero as ˙ C → 0. Thus e H is complex differentiable at C with derivative D e H ( C ) = L . (cid:3) We will apply Lemma C.1 with C = C ( q ) k , the covariance matrices which arisein the finite range decomposition (see Proposition 4.1), and B = κ B k = 2 Ch − B k where B k is as in Lemma 5.3. Now an important point is that the finite rangedecomposition in Proposition 4.1 does not yield a bound on terms likeTr (cid:0) C ( q ) k (cid:1) − D q ˙ C ( q ) k ˙ q which are independent of k and N .In order to derive bounds on the derivatives of q H ( C ( q ) k ) which are inde-pendent of k and N we now derive different expressions for the derivatives of H which do not involve C − but which require derivatives of g . This leads to a loss of 10 C. GAUSSIAN CALCULUS regularity when we consider the convolution operator g R g ( · + x ) µ C (d x ) as anoperator between function spaces and we shall see later how to deal with this lossof regularity.In the following we assume that(C.9) e , . . . , e dim ( V ) is an orthonormal basis of V. Lemma C.2 . Let B ∈ Sym ( ≥ ) ( V ) and let g ∈ C ( V ) with (C.10) sup x ∈ V X s =0 | D s g ( x ) | e − ( B x,x ) < ∞ . Furthermore, let C ∈ Sym (+) ( V ) be given with C − > B . Let ˙ C ∈ Sym (+) ( V ) anddefine h ( t ) := Z V g ( x ) µ C + t ˙ C (d x ) . Then h is a C -function on some interval ( − a , a ) and (C.11) h ′ ( t ) = Z V (cid:0) Ag (cid:1) ( x ) µ C + t ˙ C (d x ) , where (C.12) Ag ( x ) := 12 dim ( V ) X i,j =1 ˙ C i,j D g ( x, e i , e j ) , with ˙ C i,j := ( ˙ C e i , e j ) . Remark C.3 . In coordinate free notation the map A in (C.12) can be writtenas Ag ( x ) = Tr (cid:0) Hess ( g ( x )) ˙ C (cid:1) , where Hess ( g ( x )) is the linear map V → V defined by (cid:0) Hess ( g ( x )) a, b (cid:1) = D g ( x, a, b ) for all a, b ∈ V. Sometimes it is more convenient to use an orthonormal basis of the complexification e V of V to evaluate Ag . If we extend Hess ( g ( x )) as a C -linear map and D g ( x, · , · )as a C -bilinear map, then (cid:0) Hess ( g ( x )) a, b (cid:1) = D g ( x, a, b ) for all a, b ∈ e V since the sesquilinear form ( · , · ) on e V × e V is anti-linear in the second argument. Ifwe also extend ˙ C as a C -linear map and if f . . . , f dim ( V ) is an orthonormal basisof e V , thenTr V (cid:0) Hess ( g ( x )) ˙ C (cid:1) = Tr e V (cid:0) Hess ( g ( x )) ˙ C (cid:1) = dim V X i =1 (cid:0) Hess ( g ( x )) ˙ C f i , f i (cid:1) . Hence(C.13) Ag ( x ) = dim ( V ) X i =1 D g ( x, ˙ C f i , f i ) . ⋄ . GAUSSIAN CALCULUS 111 Proof. One can easily check that the definition of A is independent of thechoice of the orthonormal basis. The whole statement is invariant under isometries.Hence we may assume that V = R n with the standard scalar product and that e , . . . , e n is the standard basis. Furthermore, we write C ( t ) := C + t ˙ C in thefollowing. The starting point is the formula for the Fourier transform of a Gaussian(C.14) Z R n e − i ( ξ,x ) µ C ( t ) (d x ) = e − ( C ( t ) ξ,ξ ) . By continuity of t C ( t ) we may assume that there is an a > δ > t ∈ ( − a , a ) we have B ≤ C − ( t ) − δ Id and C ( t ) ≥ δ Id. From now onwe consider h ( t ) only on the interval ( − a , a ).Now assume first that g belong to the Schwartz class S ( R n ) of smooth andrapidly decreasing functions. By Plancherel’s formula we have(C.15) h ( t ) = Z R n g ( x ) µ C ( t ) (d x ) = 1(2 π ) n Z R n ˆ g ( ξ )e − ( C ( t ) ξ,ξ ) d ξ . Since g ∈ S ( R n ), the right hand side is differentiable with respect to t and theidentity d ∂ j g ( ξ ) = iξ j ˆ g ( ξ ) yields, with another application of Plancherel’s formula,˙ h ( t ) = − 12 1(2 π ) n Z R n ˆ g ( ξ ) n X j,k =1 ˙ C jk ξ j ξ k e − ( C ( t ) ξ,ξ ) d ξ = 12 1(2 π ) n Z R n n X j,k =1 ˙ C jk ( \ ∂ j ∂ k g ( ξ ) e − ( C ( t ) ξ,ξ ) d ξ = 12 Z R n n X j,k =1 ˙ C jk ( ∂ j ∂ k g )( x ) µ C ( t ) (d x ) = 12 Z R n Tr( ˙ C D g ( x )) µ C ( t ) (d x )= Z R n Ag ( x ) µ C + t ˙ C (d x ) . This proves assertion (C.11) and (C.12) for g ∈ S ( R n ). For a general g we usea cut-off and a convolution with a mollifier. To do so we first rewrite the result for g ∈ S ( R n ) in the integral form(C.16) Z R n g ( x ) µ C ( t ) (d x ) − Z R n g ( x ) µ C (0) (d x ) = Z t Z R n Tr( ˙ C ( s ) D g ( x )) µ C ( s ) (d x ) d s. Now, for g ∈ C ( R n ) consider the Gaussian measure h k ( x )d x on R with covariance k and define g k := h k ∗ g ∈ S ( R n ). Hence (C.16) holds for g k and we have auniform convergence g k → g and D g k → D g . Since C ( s ) ≥ δ Id we can pass tothe limit using the dominated convergence theorem which proves (C.16) whenever g ∈ C ( R n ). Finally, for g as in the lemma we let η ∈ C ∞ c ( R n ) to be a cut-offfunction that vanishes outside the unit ball B (0 , 1) and equals 1 in the ball B (0 , ).Let g k ( x ) = ϕ ( xk ) g ( x ). Then g k ∈ C ( R n ) with g k → g and D g k → D g uniformlyon compact subsets and(C.17) sup | g k ( x ) | + sup | D g k ( x ) | ≤ C sup X s =0 |∇ s g ( x ) | . Since C − ( s ) ≥ B + δ Id we may pass to the limit by the dominated convergencetheorem. This shows that (C.16) holds for all g ∈ C ( R n ) which satisfy (C.10) 12 C. GAUSSIAN CALCULUS with r = 1. Finally continuity of t C ( t ), the bound B ≤ C − ( s ) − δ Id andthe dominated convergence theorem imply that s R R n Tr( ˙ C D g ( x )) µ C ( s ) (d x ) iscontinuous. This finishes the proof. (cid:3) Lemma C.4 . Let B ∈ Sym ( ≥ ) and assume that g ∈ C ℓ ( V ) , ℓ ∈ N , satisfies (C.18) sup x ∈ V ℓ X s =0 (cid:12)(cid:12) D s g ( x ) (cid:12)(cid:12) e − ( B x,x ) < ∞ . Assume that C ∈ Sym (+) with C − > B . Then the function H defined by (C.1) satisfies (C.19) D ℓ H ( C , ˙ C , . . . , ˙ C ℓ ) = Z V (cid:0) A ˙ C · · · A ˙ C ℓ g (cid:1) ( x ) µ C (d x ) , where for f ∈ C ( V ) the operator A ˙ C i is defined by (C.20) ( A ˙ C i f )( x ) = 12 dim ( V ) X i,j =1 ˙ C i,j D f ( x, e i , e j ) . Proof. Since we already know that H is analytic in U it suffices to show theresult for ˙ C = · · · = ˙ C ℓ = ˙ C . The full result follows by polarization. It thus sufficesto show that the function h in Lemma C.2 satisfies(C.21) d k d t k h ( t ) = Z V (cid:0) A k g (cid:1) ( x ) µ C + t ˙ C (d x ) for 1 ≤ k ≤ ℓ, where A = A ˙ C . We prove this by induction. The case k = 1 is just Lemma C.2.Thus assume that k ≤ ℓ − k . Let e g := A k g . Then e g satisfiesthe assumptions of Lemma C.2. Thus by the induction assumption and Lemma C.2,we obtain d k +1 d t k +1 h ( t ) = dd t Z V e g ( x ) µ C + t ˙ C (d x ) = Z V (cid:0) A e g (cid:1) ( x ) µ C + t ˙ C (d x )= Z V (cid:0) A k +1 g (cid:1) ( x ) µ C + t ˙ C (d x ) . (cid:3) We finally collect formulae for the derivatives up to the third order for a generaldependence, that is, we now let ( − δ, δ ) ∋ t C ( t ) ∈ Sym (+) ( V ) be a C ℓ map with C (0) − > B and let g satisfies the assumptions of Lemma C.4. Then(C.22) e h ( t ) := Z V g ( x ) µ C ( t ) (d x )is a C ℓ map on some interval ( − δ ′ , δ ′ ) and the derivatives of e h can be computed bythe chain rule. In particular we obtain the following formulae.˙ e h ( t ) = DH ( C ( t ) , ˙ C ( t )) , (C.23) ¨ e h ( t ) = D H ( C ( t ) , ˙ C ( t ) , ˙ C ( t )) + DH ( C ( t ) , ¨ C ( t )) , (C.24) ... e h ( t ) = D H ( C ( t ) , ˙ C ( t ) , ˙ C ( t ) , ˙ C ( t )) + 3 D H ( C ( t ) , ˙ C ( t ) , ¨ C ( t ))(C.25) + DH ( C ( t ) , ¨ C ( t )) . . GAUSSIAN CALCULUS 113 In general D k e h ( t ) is a sum of terms of the form(C.26) D ℓ H ( C ( t ) , A , . . . , A k )with(C.27) A i = D j C ( t ) and ℓ X i =1 j i = k. PPENDIX D Chain Rules Here we formulate and prove a chain rule with loss of regularity for a composi-tion of two maps. It turns out that proving the needed claims as well as checkingtheir assumptions in particular cases is much simpler when formulated in terms ofhigher order one-dimensional directional derivatives and the related Peano deriva-tives. We first review their properties and the mutual relations. D.1. Motivation Before we enter into the precise statement of the setting and the results weconsider a simple example how loss of regularity can easily arise even for seeminglyinnocuous maps and we sketch the key calculation in the proof of the main re-sult. Consider the space C k ( S ) of 2 π -periodic k times continuously differentiablefunctions and the map F : C k ( S ) × R → C k ( S ) defined by F ( y, p )( t ) = sin( y ( t − p )) . It is easy to see that F is continuous and that the map y F ( y, p ) is smooth (infact real-analytic) as a map from C k ( S ) to itself. For a fixed y ∈ C k ( S ) \ C k +1 ( S )the map p F ( y, p ) is, however, not differentiable as a map from R to C k ( S ). Itis only differentiable as a map from R to C k − ( S ) and we have ∂∂p F ( y, p )( · ) = − cos y ( · − p ) y ′ ( · − p ) . Similarly p F ( y, p ) is a C l map to C k − l for l ≤ k . Thus each derivative with re-spect to p leads to loss of one derivative in y . A similar phenomenon occurs if we useformula (C.11) to compute the derivative of the convolution maps G ( g, C ) := g ∗ µ C with respect to the covariance C . Our renormalisation step involves a compositionof several maps of this type and one might think that this leads to a multiple loss ofregularity. The main result of this appendix, Theorem D.29 below, shows that thisis not the case. The behaviour of the composed map is no worse than the behaviourof the individual maps.To state the result informally consider scales of of Banach spaces X m ⊂ X m − ⊂ . . . ⊂ X , Y m ⊂ . . . ⊂ Y and Z m ⊂ . . . ⊂ Z as well as a Banachspace P and maps G : X m × P → Y m , F : Y m × P → Z m The present version of this Appendix is based on notes written by David Preiss. He has notonly provided a suitable framework for smoothness, in terms of classes C m ∗ and e C m introducedbelow, with particularly clear proofs of chain rule with loss of regularity, but he has also shown(Theorem D.10) that functions from C m ∗ have continuous, multilinear, and symmetric directionalderivatives. Nevertheless, all deficiencies of the present Appendix are the author’s fault. and the composed map H ( x, p ) := F ( G ( x, p ) , p ) . Informally, the assumptions on F and G are that these maps are well-behavedwith respect to the first argument, but each derivative with respect to the secondargument leads to a loss of order one in the scale of Banach spaces, i.e., that for all0 ≤ n ≤ m − l (D.1) D j D l F ( y, p ) : Y jn + l × P l → Z n is boundedand(D.2) D j D l G ( x, p ) : X jn + l × P l → Y n is bounded.Then we want to show that(D.3) D j D l H ( y, p ) : X jn + l × P l → Z n is bounded.If we assume that all natural expressions make sense this can be seen as follows.From the chain rule we deduce inductively that D l H ( x, p, ˙ p l ) := D l H ( x, p, ˙ p, . . . , ˙ p )is a weighted sum of the terms D k D i F ( G ( x, p ) , p, D l G ( x, p, ˙ p l ) , . . . , D l k G ( x, p, ˙ p l k ) , ˙ p i )with k ≥ i + P ks =1 l s = l . Another application of the chain rule shows that D j D l H ( x, p, ˙ x j , ˙ p l ) is a weighted sum of the terms D k +¯ k D i F ( G ( x, p ) , p, D ¯ j G ( x, p, ˙ x ¯ j ) , . . . , D j k D l k G ( x, p, ˙ x j k , ˙ p l k ) , ˙ p i )with ¯ j r ≥ j s ≥ l s ≥ ¯ k X r =1 ¯ j r + k X s =1 j s = j, i + k X s =1 l s = l. In particular we have l s ≤ l − i and hence D j s D l s G : X j s n + l × P l s → Y n + l − ( l − i ) = Y n + i is bounded . Moreover D k +¯ k D i F : Y k +¯ kn + i × P i → Z n is bounded . Thus k D j D l H ( x, p, ˙ x j , ˙ p l ) k Z n is bounded in terms of k ˙ x k j X n + l and k ˙ p k l P . By polar-ization we get the desired assertion (D.3). The main point in the proof of TheoremD.29 is to give a precise definition of the informal assumptions (D.1) and (D.2) andto show that under these assumptions all the operations performed above makesense. D.2. Derivatives and their relationsDirectional derivatives. Definition D.1 . Let X and Y be normed linear spaces, U ⊂ X open and G : U → Y be a function. Directional derivatives of G at x ∈ U in directions˙ x , . . . , ˙ x j ∈ X are defined by(D.4) D j G ( x, ˙ x , . . . , ˙ x j ) = dd t j . . . dd t G ( x + X t k ˙ x k ) (cid:12)(cid:12)(cid:12) t = ... = t j =0 . .2. DERIVATIVES AND THEIR RELATIONS 117 We will use the shorthand D j G ( x, ˙ x j ) = D j G ( x, ˙ x, . . . , ˙ x | {z } j ), and later, similarly, D j G ( x, ˙ x j , . . . , ˙ x j k k ) = D j G ( x, ˙ x , . . . , ˙ x | {z } j , . . . , ˙ x k , . . . , ˙ x k | {z } j k )with j = P ks =1 j s . Definition D.2 . We use C m ∗ ( U , Y ) to denote the set of continuous functions G : U → Y such that for each j ≤ m and ˙ x ∈ X , the derivative D j G ( x, ˙ x j ) existsand the map ( x, ˙ x ) ∈ U × X → D j G ( x, ˙ x j ) ∈ Y is continuous. Remark D.3 . The star ∗ is added just to indicate that this is not the standardclass C m of m -differentiable functions. Also, this definition is formally much weakerthan that by Hamilton [ Ham82 ] who takes G to be m -times differentiable if D m f : U × X × · · · × X | {z } m → Y exists and is continuous (jointly as a function on the productspace). However, Theorem D.10 below shows that it actually yields the same space.Note that for X = R it follows directly from the definition of C m ∗ ( U , Y ) that C m ∗ ( U , Y ) = C m ( U , Y ). We will see in Proposition D.17 that this identity holdswhenever X is finite dimensional. ⋄ In proofs, especially when proving chain rules, it is often useful to rely on thenotion of Peano derivatives. Definition D.4 . The Peano derivatives G ( n ) ( x, ˙ x ) of a function G at x indirection ˙ x are defined inductively by(D.5) G ( n ) ( x, ˙ x ) = n ! lim t → G ( x + t ˙ x ) − P n − j =0 G ( j ) ( x, ˙ x ) j ! t j t n whenever the derivative exists. Equivalently,(D.6) (cid:13)(cid:13)(cid:13) G ( x + t ˙ x ) − n X j =0 G ( j ) ( x, ˙ x ) j ! t j (cid:13)(cid:13)(cid:13) Y = o ( t n ) as t → Lemma D.5 . We notice the following obvious properties of these derivatives. (a) G (0) ( x, ˙ x ) exists iff G is continuous at x in direction ˙ x ; then G (0) ( x, ˙ x ) = G ( x ) . (b) G ( n ) ( x, t ˙ x ) = t n G ( n ) ( x, ˙ x ) . We show that C n ∗ ( U , Y ) can be equivalently defined using the Peano derivatives. Lemma D.6 . Suppose G is m -times Peano differentiable at every point of theline segment [ x, x + ˙ x ] in the direction of ˙ x . Then for any ≤ j ≤ n ≤ m , (cid:13)(cid:13)(cid:13) G ( j ) ( x + ˙ x, ˙ x ) − n − j X i =0 G ( j + i ) ( x, ˙ x ) i ! (cid:13)(cid:13)(cid:13) Y ≤ sup ≤ τ ≤ (cid:13)(cid:13)(cid:13) G ( n ) ( x + τ ˙ x, ˙ x ) − G ( n ) ( x, ˙ x )( n − j )! (cid:13)(cid:13)(cid:13) Y . Proof. The case j = n is obvious. When j < n , X = Y = R and ˙ x = 1, theinequality follows immediately from the mean value statement of [ Oli54 , Theorem2(ii)]. To prove the general case, find y ∗ ∈ Y ∗ realizing the norm on the left anduse the special case for the map t ∈ R → y ∗ G ( x + t ˙ x ) ∈ R . (cid:3) Proposition D.7 . G ∈ C m ∗ ( U , Y ) iff G ( n ) ( x, ˙ x ) , n ≤ m exist and are contin-uous on U × X . Moreover, for such G , D n G ( x, ˙ x n ) = G ( n ) ( x, ˙ x ) on U × X for n ≤ m . 18 D. CHAIN RULES Proof. If G ∈ C m ∗ ( U , Y ) and the segment [ x, x + ˙ x ] ⊂ U , then the function( − ǫ, ǫ ) ∋ t G ( x + t ˙ x ) ∈ Y is m -times continuously differentiable, and, inview of [ Die60 , 8.14.3 and 8.14, Problem 5],(D.7) (cid:13)(cid:13)(cid:13) G ( x + t ˙ x ) − n X j =0 D j G ( x, ˙ x j ) j ! t k (cid:13)(cid:13)(cid:13) Y = o ( t n ) as t → , for each n ≤ m , yielding G ( j ) ( x, ˙ x ) = D j G ( x, ˙ x j ), j = 0 , , . . . , m .For the opposite implication, suppose G ( m ) exists and is continuous on U × X .Given any ( x, ˙ x ) ∈ U × X , for small enough | t | we may use Lemma D.6 with n = m and t ˙ x instead of ˙ x to infer that for each 0 ≤ j < n = j + 1 ≤ m , (cid:13)(cid:13)(cid:13) G ( j ) ( x + t ˙ x, ˙ x ) − X i =0 G ( j + i ) ( x, ˙ x ) t j (cid:13)(cid:13)(cid:13) Y = o ( t ) as t → ddt G ( j ) ( x + t ˙ x, ˙ x ) (cid:12)(cid:12) t =0 = G ( j +1) ( x, ˙ x ). Hence D n G ( x, ˙ x n ) exists andequals to G ( n ) ( x, ˙ x ) for every ( x, ˙ x ) ∈ U × X and 0 ≤ n ≤ m . Since G ( n ) arecontinuous, G ∈ C m ∗ ( U , Y ). (cid:3) We also show that in the presence of continuity it suffices to require the exis-tence of the Peano derivatives in a rather weak sense. Lemma D.8 . Suppose G : U → Y and g j : U × X → Y , ≤ j ≤ m , arecontinuous functions such that for a weak ∗ dense set of y ∗ ∈ Y ∗ , y ∗ ◦ G is m -times Peano differentiable on U with its j th Peano derivative being y ∗ ◦ g j . Then G ∈ C m ∗ ( U , Y ) and D k G ( x, ˙ x j ) = G ( j ) ( x, ˙ x ) = g j ( x, ˙ x ) . Proof. For the y ∗ for which the assumption holds, Proposition D.7 showsthat y ∗ ◦ G ∈ C m ∗ ( U , R ) and D j ( y ∗ ◦ G )( x, ˙ x j ) = y ∗ ◦ g j ( x, ˙ x ). Hence, whenever thesegment [ x, x + t ˙ x ] is contained in U , y ∗ (cid:16) G ( x + t ˙ x ) − m X j =0 g j ( x, ˙ x ) j ! t j (cid:17) = 1 m ! Z t ( t − s ) m y ∗ (cid:0) g m ( x + s ˙ x, ˙ x ) − g m ( x, ˙ x ) (cid:1) d s. The function s ∈ [0 , t ] → ( t − s ) m ( g m ( x + s ˙ x, ˙ x ) − g m ( x, ˙ x )) is continuous, hence itsRiemann integral, say I , exists as an element of the completion of Y . But since bythe above y ∗ ( I ) = y ∗ (cid:0) G ( x + t ˙ x ) − P mj =0 g j ( x, ˙ x ) j ! t j (cid:1) for a weak ∗ dense set of y ∗ ∈ Y ∗ , G ( x + t ˙ x ) − m X j =0 g j ( x, ˙ x ) j ! t j = 1 m ! Z t ( t − s ) m (cid:0) g m ( x + s ˙ x, ˙ x ) − g m ( x, ˙ x ) (cid:1) d s. Since g m is continuous, G is m times Peano differentiable at every x ∈ U as amapping of U to Y , with continuous G ( j ) ( x, ˙ x ) = g j ( x, ˙ x ). So the statement followsfrom Proposition D.7. (cid:3) The previous Lemma will be used in the situation when G : U → Y and Y ֒ → V (meaning Y is a linear subspace of V and k · k V ≤ k · k Y ) to requiredifferentiability for the map G : U → V only. Corollary D.9 . Suppose Y ֒ → V and G : U → Y is m times Peano differ-entiable when considered as a map to V and such that each function G ( j ) ( x, ˙ x ) , ≤ j ≤ m , has values in Y and is continuous as a map of U × X to Y . Then G ∈ C m ∗ ( U , Y ) and D j G ( x, ˙ x j ) = G ( j ) ( x, ˙ x ) . .2. DERIVATIVES AND THEIR RELATIONS 119 Proof. Since V ∗ is weak ∗ dense in Y ∗ , Lemma D.8 is applicable with g j ( x, ˙ x ) = G ( j ) ( x, ˙ x ) . (cid:3) Multilinearity and symmetry of derivatives. Theorem D.10 . X , Y be normed linear spaces with U ⊂ X open, and let G ∈ C m ∗ ( U , Y ) . Then, for every ≤ j ≤ m , the directional derivative D j G ( x, ˙ x , . . . , ˙ x j ) exists for all x ∈ U and ˙ x , . . . , ˙ x j ∈ X .Moreover, it is a continuous, symmetric, j -linear map in the variables ˙ x , . . . , ˙ x j and D j G ∈ C m − j ∗ ( U × X j , Y ) . The main idea is to get information on the map s G ( j ) ( x + sv, ˙ x, . . . , ˙ x ) bywriting G ( x + s ( v + t ˙ x )) = G ( x + sv + st ˙ x )and using Peano differentiability of G at x on the left hand side and Peano differ-entiability at x + sv on the right hand side. A key tool is the following polynomialinterpolation lemma. Theorem D.10 will then be a consequence of Proposition D.12below. Lemma D.11 . For any j = 0 , . . . , m , let Φ j : ( − s , s ) → X be bounded and Ψ j : R → X . Suppose that (D.8) m X j =0 s j ( Ψ j ( t ) − Φ j ( s ) t j ) = o ( s m ) as s → for every t ∈ R . Then for each j = 0 , . . . , m : (a) The function Ψ j is a polynomial of degree at most j and (b) there exists a polynomial p j : R → X of degree at most m − j such that Φ j ( s ) = p j ( s ) + o ( s m − j ) as s → . (c) Moreover, if b Φ j , b Ψ j also satisfy (D.8) then k b Φ j − Φ j k poly ≤ C lim sup s → sup t ∈ (0 , (cid:13)(cid:13)(cid:13) m X j =0 s j ( b Ψ j − Ψ j ( t )) (cid:13)(cid:13)(cid:13) . Proof. Fix different t , . . . , t m ∈ (0 , 1) and let q j be the corresponding La-grange basis polynomials, q j ( t k ) = δ k,j . Then for every t ∈ R ,(D.9) m X j =0 s j (cid:16) Ψ j ( t ) − m X k =0 Ψ j ( t k ) q k ( t ) (cid:17) == m X j =0 s j ( Ψ j ( t ) − Φ j ( s ) t j ) − m X k =0 q k ( t ) m X j =0 s j ( Ψ j ( t k ) − Φ j ( s ) t jk ) = o ( s m ) , implying that Ψ j ( t ) − P mk =0 Ψ j ( t k ) q k ( t ) = 0 for each j = 0 , , . . . , m and thus each Ψ j ( t ) is a polynomial of degree at most m . Only now we use that Φ j are bounded,yielding from (D.8) that P jk =0 s k ( Ψ k ( t ) − Φ k ( s ) t k ) = o ( s j ) for every j = 0 , . . . , m ,and the above argument with j instead of m shows that Ψ j has degree at most j . For p ( s ) = P nℓ =0 p ℓ s ℓ we define k p k poly = max ℓ =0 ,...,n | p ℓ | . 20 D. CHAIN RULES For (b), let 0 ≤ ℓ ≤ m and find a k so that P mi =0 a k t jk = δ j,ℓ . By the degreeestimate on Ψ j , P mk =0 a k Ψ j ( t k ) = 0 for j < ℓ . Hence(D.10) Φ ℓ ( s ) − m − ℓ X j =0 s j m X k =0 a k Ψ j + ℓ ( t k ) = − s − ℓ m X k =0 a k m X j =0 s j ( Φ j ( s ) t jk − Ψ j ( t k )) = o ( s m − ℓ ) . For (c), we just notice that, in view of (D.10), the coefficients of p k ( s ) are linearcombinations (with fixed coefficients) of the values Ψ j + k ( t k ) with t k ∈ (0 , (cid:3) Proposition D.12 . Let G ∈ C m ∗ ( U , Y ) . Then for every ≤ j ≤ m , thedirectional derivative D j G ( x, ˙ x , . . . , ˙ x j ) exists for all x ∈ U and ˙ x , . . . , ˙ x j ∈ X , itis symmetric and j -linear in the variables ˙ x , . . . , ˙ x j , and D j G ∈ C m − j ∗ ( U × X j , Y ) . Proof. We show that f ( x, ˙ x ) := G (1) ( x, ˙ x ) belongs to C m − ∗ ( U × X , Y ) and islinear in ˙ x . Used recursively, this shows that for each 1 ≤ j ≤ m , ( x, ˙ x , . . . , ˙ x j ) → D j G ( x, ˙ x , . . . , ˙ x j ) is j -linear in ˙ x , . . . , ˙ x j and belongs to C m − j ∗ ( U × X j , Y ). Re-call that by Proposition D.7, G is m -times Peano differentiable and G ( j ) ( x, ˙ x ) = D j G ( x, ˙ x j ) for j ≤ m , x ∈ U , and ˙ x ∈ X .Fix x, ˙ x, v ∈ X and denote Φ j ( s ) = G ( j ) ( x + sv, ˙ x ) /j ! and Ψ j ( t ) = G ( j ) ( x, v + t ˙ x ) /j !. By definition, for each t ∈ R , G ( x + s ( v + t ˙ x )) = P mj =0 Ψ j ( t ) s j + o ( s m ).Also, by Lemma D.6,(D.11) k G (( x + sv ) + st ˙ x ) − m X j =0 Φ j ( s )( st ) j k ≤≤ ( st ) m sup ≤ τ ≤ k G ( m ) ( x + sv + τ st ˙ x, u ) − G ( m ) ( x + sv ) k = o ( s m ) . Hence P mj =0 s j ( Ψ j ( t ) − Φ j ( s ) t j ) = o ( s m ) and we see from Lemma D.11(a) that G (1) ( x, v + t ˙ x ) = a + bt for some a, b . For t = 0 we get a = G (1) ( x, v ) andby continuity, b = lim t →∞ G (1) ( x, v/t + ˙ x ) = G (1) ( x, ˙ x ). Hence G (1) ( x, v + ˙ x ) = G (1) ( x, v ) + G (1) ( x, ˙ x ), and we infer that f ( x, ˙ x ) = G (1) ( x, ˙ x ) is linear in the secondvariable.By Lemma D.11(b), for each fixed x, ˙ x the function g ˙ x ( x ) = f ( x, ˙ x ) has thePeano derivative g ( j )˙ x ( x, v ), j = 1 , . . . , m − 1. Moreover, continuity of Peano deriva-tives G ( n ) and Lemma D.11(c) imply that ( x, ˙ x, v ) → g ( j )˙ x ( x, v ) is continuous on U × X . Since f ( x, ˙ x ) is linear in ˙ x ,(D.12) f (( x, ˙ x ) + t ( u, ˙ u )) − f (( x, ˙ x )) = g ˙ x ( x + tu ) − g ˙ x ( x ) + tg ˙ u ( x + tu ) , showing that f is m − f belongsto C m − ∗ ( U × X , Y ) by Proposition D.7.Symmetry of the directional derivatives follows from the following lemma. (cid:3) Lemma D.13 . Let G : U → Y and fix (not necessarily distinct) ˙ x , . . . , ˙ x k ∈ X .Suppose that the directional derivative x ∈ U → D j G ( x, ˙ x j , . . . , ˙ x j k k ) exists and iscontinuous whenever j := j + · · · + j k ≤ m . Then for any t , . . . , t k ∈ R , (D.13) G ( j ) ( x, k X s =1 t s ˙ x s ) = j ! X j + ··· + j k = j D j G ( x, ˙ x j , . . . , ˙ x j k k ) t j . . . t j k k j ! · · · j k ! . .2. DERIVATIVES AND THEIR RELATIONS 121 In particular, D k G ( x, ( P ks =1 t s ˙ x s ) k ) = G ( k ) ( x, P ks =1 t s ˙ x s ) exists and (D.14) D k G ( x, ˙ x , . . . , ˙ x k ) = D k G ( x, ˙ x π (1) , . . . , ˙ x π ( k ) ) for every permutation π of { , . . . , k } . Proof. Expanding recursively and estimating errors by Lemma D.6, we get(D.15) G ( x + t X t s ˙ x s ) = X j := j + ··· + j k ≤ m D j G ( x, ˙ x j , . . . , ˙ x j k k ) t j . . . t j k k j ! · · · j k ! t j + o ( t m ) , which shows (D.13). Since the right hand side of (D.13) is continuous in x , Propo-sition D.7 used separately on each line in the direction P ks =1 t s ˙ x s implies that theiterated derivative D k G ( x, ( P ks =1 t s ˙ x s ) k ) exists and equals G ( k ) ( x, P ks =1 t s ˙ x s ).Using the equality (D.13) with P ks =1 t s ˙ x s replaced by P ks =1 t π ( s ) ˙ x π ( s ) gives thesame left hand side. Since the right side is a polynomial, the coefficients in front of t · · · t k are equal, giving the last statement. (cid:3) Remark D.14 . Notice that the order of directions in the recursive expansioncan be chosen. As a result, the assumption can be narrowed, say in the case oftwo directions { ˙ x , ˙ x } , to the assumption that the directional derivative x ∈ U → D j G ( x, ˙ x j , ˙ x j , ˙ x j ) exists and is continuous whenever j := j + j + j ≤ m and j ∈ { , } . ⋄ The following Corollary is a useful criterion for proving that a given func-tion on a product space belongs to C m ∗ . It involves partial derivatives which aredefined and denoted in the standard way. In particular, D j D ℓ G (( x, p ) , ˙ p ℓ , ˙ x j ) = D j + ℓ G (( x, p ) , (0 , ˙ p ) ℓ , ( ˙ x, j ). Corollary D.15 . Suppose G : O ⊂ X × P → Y , m ∈ N , and for each j + ℓ ≤ m , the derivative ( x, p, ˙ x, ˙ p ) → D j D ℓ G (( x, p ) , ˙ p ℓ , ˙ x j ) exists and is continuouson O × X × P . Then G ∈ C m ∗ ( O , Y ) . Proof. Lemma D.13 shows that for each j ≤ m the Peano derivative G ( j ) (( x, p ) , ( ˙ x, ˙ p )) = D j G (( x, p ) , (( ˙ x, 0) + (0 , ˙ p )) j ) == j X k =0 (cid:18) jk (cid:19) D j G (( x, p ) , (0 , ˙ p ) k , ( ˙ x, j − k ) == j X k =0 (cid:18) jk (cid:19) D j − k D k G (( x, p ) , ˙ p k , ˙ x j − k )exists and is continuous. Hence G ∈ C m ∗ ( O , Y ) by Proposition D.7. (cid:3) Remark D.16 . Notice that in view of Remark D.14, there is also a flexibilityin the demanded order of partial derivatives in the condition in the Corollary. ⋄ Relation to usual derivatives. Proposition D.17 . Using C m ( U , Y ) to denote the usual spaces of Fr´echetdifferentiable functions (with operator norms on multilinear forms from L m ( X , Y ) )and m ≥ , we have C m ( U , Y ) = (cid:8) G ∈ C m ∗ ( U , Y ) : D m G ∈ C ( U , L m ( X , Y )) (cid:9) ⊃ C m +1 ∗ ( U , Y ) . If X is finite dimensional then C m ( U , Y ) = C m ∗ ( U , Y ) . 22 D. CHAIN RULES Proof. We first show the inclusion(D.16) { G ∈ C m ∗ ( U , Y ) : D m G ∈ C ( U , L m ( X , Y ) } ⊃ C m +1 ∗ . Let G ∈ C m +1 ∗ ( U , Y ). Given x ∈ U find δ > k D m +1 G ( x + ˙ x, ˙ x , . . . , ˙ x m +1 ) k ≤ {k ˙ x k , k ˙ x i k} ≤ δ. Hence for k ˙ x k < εδ m +1 and max i k ˙ x i k ≤ k D m G ( x + ˙ x, ˙ x , . . . , ˙ x m ) − D m G ( x, ˙ x , . . . , ˙ x m ) k == δ − m k D m G ( x + ˙ x, δ ˙ x , . . . , δ ˙ x m ) − D m G ( x, δ ˙ x , . . . , δ ˙ x m ) k ≤≤ δ − m − sup Here we consider the chain rule showing that F ◦ G ∈ C m ∗ ( U , Z ) in the sit-uation when G : U → Y , F : Y → Z , where U and Y are open subsets of X and Y , respectively, and G ∈ C m ∗ ( U , V ) for some Y ֒ → V (meaning, as above,that Y is a linear subspace of V and k·k V ≤ k·k Y ). This generalizes the chainrule of [ Ham82 , Theorem 3.6.4] where V = Y and F is assumed to belong to C m ∗ ( Y , Z ). In our situation, although F ◦ G obviously makes sense, expressionssuch as DF ( G ( x ) , DG ( x, ˙ x )) may not, since derivatives of G belong to V and sonot to the domain of the derivative of F . So for the chain rule to hold, a naturalassumptions are that Y is dense in V and D j F has a continuous extension from Y × Y j to Y × V j . (The density of Y in V is not really needed, but is conve-nient since it guarantees that the extension is unique and j -multilinear in the lastvariables.) Definition D.18 . We use C m V ( Y , Z ) to denote the space of maps F : Y ⊂ Y → Z such that for any j ≤ m , the derivative D j F exists and can be extended toa continuous map D j V F of Y × V j to Z (with a slight abuse of notation we usuallyskip the subscript V from D j V ). Remark D.19 . (a) For j = 0 this requires only that F : Y → Z be continuous.(b) Proposition D.7 and the polarization formula show that it suffices to extendthe maps ( y, ˙ y ) ∈ Y × Y → D j F ( y, ˙ y j ) to continuous maps defined on Y × V .(c) By Proposition D.7, C m V ( Y , Z ) ⊂ C m ∗ ( Y , Z ) with equality when V = Y . ⋄ Lemma D.20 . Let F ∈ C m V ( Y , Z ) and j ≤ m . Then D jV F ∈ C m − j V j +1 ( Y × V j , Z ) . Proof. By the polarization formula it suffices to show that ( y, v ) → Φ ( y, v ) := D j V F ( y, v j ) belongs to C m − j V ( Y × V , Z ). Considering first Φ as a map of Y × Y to Z and using multilinearity of the derivative, we have(D.24) D k D ℓ Φ (( y, v ) , ˙ v ℓ , ˙ y k ) = j · · · ( j − ℓ + 1) D j + k F ( y, v j − ℓ , ˙ v ℓ , ˙ y k )for ℓ ≤ j and k ≤ m − j . Since these derivatives are zero for ℓ > j , we have Φ ∈ C m − j ∗ ( Y × Y , Z ) by Corollary D.15 and Theorem D.10. Moreover, expressing D s Φ ,0 ≤ s ≤ m − j , with the help of partial derivatives, we see that these derivatives havecontinuous extensions to maps ( Y × V ) × ( V × V ) s → Z implying the statement. (cid:3) Theorem D.21 . Suppose U ⊂ X and Y ⊂ Y are open, Y ֒ → V , G : U → Y , G ( U ) ⊂ Y , G ∈ C m ∗ ( U , V ) , and F : Y → Z , F ∈ C m V ( Y , Z ) . Then F ◦ G ∈ C m ∗ ( U , Z ) and D j ( F ◦ G )( x, ˙ x j ) is a linear combination of terms (D.25) D kV F ( G ( x ) , D j G ( x, ˙ x j ) , . . . , D j k G ( x, ˙ x j k )) where j s ≥ and P ks =1 j s = j . Proof. We will show existence and continuity of Peano derivatives of F ◦ G .Let x ∈ U , ˙ x ∈ X . For any t , working just on the segment I t := [ G ( x ) , G ( x + t ˙ x )] ⊂ Y 24 D. CHAIN RULES we have an estimate(D.26) (cid:13)(cid:13)(cid:13) F ( G ( x + t ˙ x )) − j X s =0 D s F ( G ( x ) , ( G ( x + t ˙ x ) − G ( x )) s ) s ! (cid:13)(cid:13)(cid:13) ≤ sup y ∈ I t (cid:13)(cid:13)(cid:13) D j F ( y, ( G ( x + t ˙ x ) − G ( x )) j ) − D j F ( x, ( G ( x + t ˙ x ) − G ( x )) j ) j ! (cid:13)(cid:13)(cid:13) for any j ≤ m . Here all derivatives of F are applied to elements of Y , so theextension has not been used yet. Since ( G ( x + t ˙ x ) − G ( x )) /t converge, in the norm k · k V , to G ′ ( x, ˙ x ), G ′ ( x, ˙ x ) ∈ V and, using continuity of the extended D j F , D j F ( y t , (( G ( x + t ˙ x ) − G ( x )) /t ) j ) → D j F ( x, G ′ ( x, ˙ x ) j ) as t → y t ∈ I t . Hence the right side of (D.26) is o ( t j ). Since x, ˙ x are fixed,expanding D s F ( G ( x ) , ( G ( x + t ˙ x ) − G ( x )) s ) is standard: D s F ( y, ˙ y , . . . , ˙ y s ) hasbeen extended to a continuous s -linear form on V s , into which one plugs a C j function R → Y ⊂ V , namely t → G ( x + t ˙ x ) − G ( x ).It follows that F ◦ G is m -times Peano differentiable with derivatives given by theterms from the expansion of D s F ( G ( x ) , ( G ( x + t ˙ x ) − G ( x )) s ), giving (D.25). Theseformulas show that ( F ◦ G ) ( s ) is continuous as a map U × X → Z . Consequently, F ◦ G ∈ C m ∗ ( U , Z ) by Proposition D.7. (cid:3) D.4. Chain rule with parameter and a graded loss of regularity In the chain rule of this section, the main point is that the inner and/or outerfunction depend on an additional parameter, the regularity of partial derivatives de-pends on the order of the derivative with respect to the parameter, and the resultingcomposition has the same regularity properties as the functions we are composing.In principle, this chain rule is very different from the one in Theorem D.29, althoughwe will reduce its proof to is. Proposition D.22 . Suppose P , Q , Y , V are normed linear spaces, P , Q and Y are open subsets of P , Q and Y , respectively, Y = Y m ֒ → Y m − ֒ → . . . ֒ → Y , Φ : P → Y and F : Y × Q → V are such that Φ ( P ) ⊂ Y and for each ≤ ℓ ≤ m , (i) Φ ∈ C m − ℓ ∗ ( P , Y ℓ ) ; (ii) for each j ≤ m − ℓ , D j D ℓ F exists on Y × Q × Q ℓ × Y j and has a continuousextension to Y × Q × Q ℓ × Y jℓ .Then the map Ψ ( p, q ) := F ( Φ ( p ) , q ) belongs to C m ∗ ( P ×Q , V ) and for each j + ℓ ≤ m the derivative D j D ℓ Ψ (( p, q ) , ˙ q ℓ , ˙ p j ) is a combination of terms (D.27) D k D ℓ F (( Φ ( p ) , q ) , ˙ q ℓ , D j Φ ( p, ˙ p j ) , . . . , D j k Φ ( p, ˙ p j k )) where j s ≥ , P ks =1 j s = j and D i D ℓ F denotes the extension from (ii) . Proof. Clearly, D ℓ Ψ (( p, q ) , ˙ q ℓ ) = D ℓ F (( Φ ( p ) , q ) , ˙ q ℓ ) exists for each 0 ≤ ℓ ≤ m ,and with fixed q and ˙ q it is a composition f q, ˙ q ◦ Φ , where f q, ˙ q ( y ) = D ℓ F (( y, q ) , ˙ q ℓ ).By (i), Φ ∈ C m − ℓ ∗ ( P , Y ℓ ), and by (ii), f q, ˙ q ∈ C m − ℓ Y ℓ ( Y , V ). Hence by Theorem D.21,the function p → D ℓ Ψ (( p, q ) , ˙ q ℓ ) belongs to C m − ℓ ∗ ( P , V ) and its j th derivative is acombination of the terms specified in (D.27).It remains to observe that ( p, q ) → (cid:0) ( Φ ( p ) , q ) , ˙ q, D j Φ ( p, ˙ p j ) , . . . , D j k Φ ( p, ˙ p j k ) (cid:1) maps, by the condition j s ≤ j ≤ m − ℓ and (i), P × Q continuously to ( Y × Q ) × Q × Y kℓ and this space is mapped by (cid:0) ( y, q ) , ˙ q, ˙ y , . . . , ˙ y k (cid:1) → D i D ℓ F (cid:0) ( y, q ) , ˙ q ℓ , ˙ y , . . . , ˙ y k (cid:1) .4. CHAIN RULE WITH PARAMETER AND A GRADED LOSS OF REGULARITY 125 continuously to V by (ii). Hence each of the functions in (D.27) maps P × Q con-tinuously to V , implying that Ψ ∈ C m ∗ ( P × Q , V ). (cid:3) Corollary D.23 . If, under the assumptions of Proposition D.22 we are alsogiven a function Υ ∈ C m ∗ ( P , Q ) with Υ ( P ) ⊂ Q , the map Θ ( p ) := F ( Φ ( p ) , Υ ( p )) be-longs to C m ∗ ( P , V ) and for each n ≤ m , the derivative D n Θ ( p, ˙ p n ) is a combinationof terms D i D k F (cid:0) ( Φ ( p ) , Υ ( p )) , D j Υ ( p, ˙ p j ) , . . . , D j i Υ ( p, ˙ p j i ) , D ℓ Φ ( p, ˙ p ℓ ) , . . . , D ℓ k Φ ( p, ˙ p ℓ k ) (cid:1) where j s , ℓ s ≥ and P is =1 j s + P ks =1 ℓ s = n . Proof. Observe that Θ = Ψ ◦ κ where Ψ comes from Proposition D.22 and κ : P → P × Q is κ ( p ) = ( p, Υ ( p )). Since κ ∈ C m ∗ ( P , P × Q ), κ ( P ) ⊂ P × Q and Ψ ∈ C m ∗ ( P × Q , V ), the statement follows from Theorem D.21. (cid:3) The following main chain rule is a ‘symmetric’ version of the above, whichis capable of being iterated. It will be stated in the following situation. Let P , X = X m ֒ → . . . ֒ → X , Y = Y m ֒ → . . . ֒ → Y and Z = Z m ֒ → . . . ֒ → Z benormed linear spaces, U ⊂ X , V ⊂ P , and Y ⊂ Y are open. We will use f X n todenote the closure of X in X n , and similarly for e Y n and e Z n . Also, we use X (andsimilarly Y and Z ) for the sequence ( X m , . . . , X ).The class of functions we will consider may be informally described as those G : U × V → Y for which D j D ℓ G is a continuous map U × V × P ℓ × f X jn → Y n + ℓ ,i.e., ℓ derivatives in the parameter p ∈ V lead to a loss of regularity of order ℓ inthe scale of Banach spaces. Since this description has several interpretations, wegive a rather detailed one as a formal definition. Definition D.24 . For any 0 ≤ k ≤ m , we define e C k ( U × V , X , Y ) as the set ofall maps G : U × V → Y such that(a) G ∈ C k ∗ ( U × V , Y ).(b) For each j + ℓ ≤ k , the function( x, p, ˙ x , . . . , , ˙ x j , ˙ p , . . . , , ˙ p ℓ ) → D j D ℓ G (( x, p ) , ˙ p , . . . , , ˙ p ℓ , ˙ x , . . . , , ˙ x j ) , which is by (a) defined as a map U × V × X j × P ℓ → Y has a (necessarilyunique) extension to a continuous mapping U × V × f X jℓ × P ℓ → Y . Thisextension is also denoted D j D ℓ G .(c) For each 0 ≤ j ≤ k − ℓ and each 0 ≤ n ≤ m − ℓ the restriction of D j D ℓ G (whichhas been already extended by (b)) to U × V × f X jn + ℓ × P ℓ has values in Y n andis continuous as a mapping between these spaces.Notice that, clearly, e C i ( U × V , X , Y )) ⊂ e C k ( U × V , X , Y ) for k ≤ i . For provingthat G ∈ e C k ( U × V , X , Y ) the following simplification of this definition is ratheruseful. Lemma D.25 . Assume that ≤ k ≤ m . Then G : U × V → Y belongs to e C k ( U × V , X , Y ) iff (i) as a map of U × V to Y , G has derivatives D j D ℓ G (( x, p ) , ˙ p ℓ , ˙ x j ) for all j + ℓ ≤ k , ( x, p ) ∈ U × V , ˙ p ∈ P and ˙ x ∈ X ; 26 D. CHAIN RULES (ii) for ≤ j ≤ k − ℓ and all ≤ n ≤ m − ℓ there is continuous map Ψ j,ℓ,n : U ×V × f X n + ℓ × P → Y n such that D j D ℓ G (( x, p ) , ˙ p ℓ , ˙ x j ) = Ψ j,ℓ,n ( x, p, ˙ x, ˙ p ) forevery ( x, p ) ∈ U × V , ˙ p ∈ P , and ˙ x ∈ X . Proof. If G ∈ e C k ( U × V , X , Y ), (i) and (ii) are obvious. For the oppositeimplication, assuming (i) and (ii) we see that for each j + ℓ ≤ k , ( x, p, ˙ x, ˙ p ) → D j D ℓ G (( x, p ) , ˙ p ℓ , ˙ x j ) is a continuous map U × V × X × P → Y . Hence G ∈ C k ∗ ( U ×V , Y ) by Corollary D.15, yielding D.24(a). Lemma D.13 and the polarizationformula establish the function( x, p, ˙ x , . . . , ˙ x j , ˙ p , . . . , ˙ p ℓ ) → D j D ℓ G (( x, p ) , ˙ p , . . . , ˙ p ℓ , ˙ x , . . . , ˙ x j )as a combination of terms( x, p, ˙ x , . . . , ˙ x j , ˙ p , . . . , ˙ p ℓ ) → D j D ℓ G (( x, p ) , ( X k ∈ I σ k ˙ p k ) ℓ , ( X k ∈ J τ k ˙ x k ) j )where I ⊂ { , . . . , ℓ } , J ⊂ { , . . . , j } , and σ k , τ k = ± 1. This shows that for each0 ≤ n ≤ m − ℓ , the derivative D j D ℓ G can be extended to a continuous map e Ψ j,ℓ,n ,from U × V × f X jn + ℓ × P ℓ to Y n . With n = 0 this shows D.24(b). For 0 ≤ n ≤ m − ℓ we see from X = X m ֒ → X n + ℓ ֒ → X ℓ that both e Ψ j,ℓ,n and the restriction of e Ψ j,ℓ, to U × V × f X jn + ℓ × P ℓ are continuous as maps of U := (cid:0) U × V × f X jn + ℓ × P ℓ , k · k X ℓ (cid:1) to Y . Since X is dense in ( f X n + ℓ , k · k X n + ℓ ), and so also in ( f X n + ℓ , k · k X ℓ ), themaps e Ψ j,ℓ,n and e Ψ j,ℓ, coincide on a dense subset of U , hence on all of U , provingD.24(c). (cid:3) Remark D.26 . Clearly, the claim remains true if one replaces D j D ℓ G (( x, p ) , ˙ p ℓ , ˙ x j )with the derivatives taken in the opposite order (see Remark D.16). In the presentand the following appendices, in the notation e C m ( U ×V , X , Y ) we indicate, somehowpedantically but usefully for clarity in proofs, the sequences X , Y of Banach spaces.When using this notion in particular applications, the sequences X and Y willbe clear from the context and we will skip them from the notation writing just e C m ( U × V ). ⋄ For working with functions from e C m ( U × V , X , Y ) it is useful to know that theyhave properties stronger than those given in the definition. Lemma D.27 . Let G ∈ e C m ( U × V , X , Y ) and ≤ j, n ≤ m − ℓ . Then (1) for fixed x ∈ U , the map p → G ( x, p ) belongs to C ℓ ∗ ( V , e Y m − ℓ ) ; (2) for fixed p ∈ V and ˙ p , . . . , ˙ p ℓ ∈ P , the (extended) map ( x, ˙ x , . . . , ˙ x j ) → D j D ℓ G (( x, p ) , ˙ p , . . . , ˙ p ℓ , ˙ x , . . . , ˙ x j ) belongs to C m − ℓ − j X j +1 n + ℓ ( U × f X jn + ℓ , e Y n ) . Proof. (1) By Corollary D.9 and D.24(c) with n = m − ℓ , the map p → G ( x, p )belongs to C ℓ ∗ ( P , Y m − ℓ ). Hence the derivative D ℓ G is an iterated limit of elementsof Y taken in the norm of Y m − ℓ , and so it belongs to e Y m − ℓ .(2) By Lemma D.20 it suffices to show that the function x → D ℓ G (( x, p ) , ˙ p , . . . , ˙ p ℓ ) .4. CHAIN RULE WITH PARAMETER AND A GRADED LOSS OF REGULARITY 127 belongs to C m − ℓ X n + ℓ ( U , e Y n ). But this follows by the same argument as in the proof of(1). (cid:3) Remark D.28 . Since (2) puts the values of the (extended) derivatives into thecorresponding closures of Y , G belongs to C m ( U × V , X , Y ) iff and only if it belongsto this space when X n and Y n are replaced by f X n and e Y n , respectively. So, atleast in proofs, we may always assume that X is dense in X n and Y in Y n . ⋄ Theorem D.29 . Let G ∈ e C m ( U ×V , X , Y ) , G ( U ×V ) ⊂ Y , F ∈ e C m ( Y ×V , Y , Z ) and define F ⋄ G : U × V → Z by F ⋄ G ( x, p ) := F ( G ( x, p ) , p ) . Then F ⋄ G ∈ e C m ( U × V , X , Z ) . Proof. By Remark D.28, we may assume f X n = X n , and similarly for Y n and Z n . Set H := F ⋄ G . For fixed x ∈ U , the function p → H ( x, p ) is of the form of acomposition F ( Φ ( p ) , Υ ( p )) where the outer function F : Y × V → Z and the innerfunctions Φ ( p ) = G ( x, p ) and Υ ( p ) = p satisfy the assumptions of Corollary D.23with Q = P , Q = P and V = Z . Hence p → H ( x, p ) belongs to C m ∗ ( P , Z ) andfor each ℓ ≤ m , the derivative D ℓ H (( x, p ) , ˙ p ℓ ) is a combination of terms(D.28) D k D i F (cid:0) ( G ( x, p ) , p ) , ˙ p i , D m G (( x, p ) , ˙ p m ) , . . . , D m k G (( x, p ) , ˙ p m k ) (cid:1) where m s ≥ i + P ks =1 m s = ℓ .We now fix p, ˙ p and differentiate the function in (D.28) with respect to x . Weset K ( x ) := (cid:0) G ( x, p ) , D m G (( x, p ) , ˙ p m ) , . . . , D m k G (( x, p ) , ˙ p m k ) (cid:1) and L ( y, ˙ y , . . . , ˙ y k ) = D k D i F (cid:0) ( y, p ) , ˙ p i , ˙ y , . . . , ˙ y k (cid:1) . Then the expression in (D.28) is given by the composition ( L ◦ K )( x ). Since m s ≤ l − i ≤ m − i we have m − m s ≥ i and it follows from Lemma D.27 (2) (applied tothe s -th component of K with n = m − m s ) that K ∈ C m − l ∗ ( U ; Y × Y ki ) . Application of Lemma D.27 (2) to F yields that L ∈ C m − i − k Y k +1 i ( Y × Y ki , Z ) ⊂ C m − ℓ Y k +1 i ( Y × Y ki , Z ) . where the inclusion follows from the relation l ≥ i + k . Hence, Theorem D.21 shows L ◦ K ∈ C m − l ∗ ( U , Z ) and for each j ≤ m − ℓ the derivative of D j ( L ◦ K ) (andhence the derivative D j D ℓ H ) exists and is given by a sum of terms of the form(D.29) D k D i F (cid:16) ( G ( x, p ) , p ) , ˙ p i , D j D ℓ G (( x, p ) , ˙ p ℓ , ˙ x j ) , . . . , D j k D ℓ k G (( x, p ) , ˙ p ℓ k , ˙ x j k ) (cid:17) where j s + ℓ s ≥ i + P ks =1 ℓ s = ℓ and P ks =1 j s = j .Finally, we rely on Lemma D.27 once more. For any s = 1 , . . . , k , the map( x, p, ˙ x, ˙ p ) → D j s D ℓ s G (( x, p ) , ˙ p ℓ s , ˙ x j s ) is a continuous map from U ×V × X n s + ℓ s × P to Y n s whenever n s ≤ m − ℓ s . Choosing n s = n + ℓ − ℓ s for any fixed n ≤ m − ℓ , weget a map U × V × X n + ℓ × P → Y n + ℓ − ℓ s . Using that ℓ s ≤ ℓ − i , the derivatives havebeen extended so that the function of ( x, p, ˙ x, ˙ p ) defined in (D.29) is a compositionof continuous maps U × V × X n + ℓ × P → Y × P × P i × Y kn + i 28 D. CHAIN RULES and Y × P × P i × Y kn + i → Z n . Hence ( x, p, ˙ x, ˙ p ) → D j D ℓ H (( x, p ) , ˙ p ℓ , ˙ x j ) is continuous as a map of U ×V × X n + ℓ × P to Z n and we conclude from Lemma D.25 that H ∈ e C m ( U × V , X , Z ). (cid:3) Remark D.30 . Let p ∈ V and assume that G ( U × B δ ( p )) ⊂ Y ,(D.30) k D j D ℓ G (( x, p ) , ˙ p ℓ , ˙ x j ) k Y n ≤ C k ˙ x k j X n + ℓ k ˙ p k ℓ for any ( x, p, ˙ x, ˙ p ) ∈ U × B δ ( p ) × X n + ℓ × P and any 0 ≤ j + ℓ ≤ m, ≤ n ≤ m − l and(D.31) k D j D ℓ F (( y, p ) , ˙ p ℓ , ˙ y j ) k Z n ≤ C k ˙ y k j Y n + ℓ k ˙ p k ℓ for any ( y, p, ˙ y, ˙ p ) ∈ Y × B δ ( p ) × Y n + ℓ × P and any 0 ≤ j + ℓ ≤ m, ≤ n ≤ m − l .Then(D.32) k D j D ℓ H (( x, p ) , ˙ p ℓ , ˙ x j ) k Z n ≤ C k ˙ x k j X n + ℓ k ˙ p k ℓ for any ( x, p, ˙ x, ˙ p ) ∈ U × B δ ( p ) × X n + ℓ × P and any 0 ≤ j + ℓ ≤ m, ≤ n ≤ m − l ,where C depends only on C , C and m . In fact, since D j D ℓ H (( x, p ) , ˙ p ℓ , ˙ x j ) is aweighted sum of the terms in (D.29) it is easy to see that there exists a constant C ( m ) such that C ≤ C ( m ) C (1 + C m ). ⋄ If we the introduce the norm k G k e C m ( U×V , X , Y ) := inf n M : k D j D ℓ G (( x, p ) , ˙ p ℓ , ˙ x j ) k Y n ≤ M k ˙ x k j X n + ℓ k ˙ p k ℓ , (D.33) ∀ ( x, p, ˙ x, ˙ p ) ∈ U × V × X n + ℓ × P and any 0 ≤ j + ℓ ≤ m, ≤ n ≤ m − l } then the remark implies that k H k can be controlled in terms of k F k and k G k . D.5. A special case of a function G that is linear in its first argument Here we discuss conditions assuring that G ∈ e C m in a special case of lineardependence on the first variable: Lemma D.31 . Let G : X × V → Y and assume that:(i) For any p ∈ V , the map x G ( x, p ) is linear.(ii) For any ≤ ℓ ≤ m and any x ∈ X , the map p G ( x, p ) is in C ℓ ∗ ( V , Y m − ℓ ) .(iii) For any p ∈ V there exists δ, C > such that k D ℓ G (( x, p ) , ˙ p ℓ ) k Y n ≤ C k x k X n + ℓ k ˙ p k ℓ for any ≤ ℓ ≤ m , ≤ n ≤ m − ℓ , and ( x, p, ˙ p ) ∈ X × B δ ( p ) × P .Then G ∈ e C m ( X × V , X , Y ) . Moreover (D.34) k G k e C m ( B R ×V , X , Y ) ≤ C ( m )(1 + R ) M ′ , where M ′ := inf (cid:8) M : k D ℓ G (( x, p ) , ˙ p ℓ ) k Y n ≤ M k ˙ x k X n + ℓ k ˙ p k ℓ , for any ( x, p, ˙ x, ˙ p ) ∈ X × V × P and any ≤ n + ℓ ≤ m } (D.35) .6. FUNCTION NOT DEPENDING ON THE PARAMETER 129 Proof. We will verify the conditions of Lemma D.25.The conditions (i) and (ii) above imply the condition Lemma D.25(i). In-deed, taking into account the linearity of G in the first variable, the derivative D G (( x, p ) , ˙ x ) exists and equals G ( ˙ x, p ) (with any norm k·k Y n , 0 ≤ n ≤ m (inparticular, also n = m − ℓ ) on the target space Y ). Thus D ℓ D G (( x, p ) , ˙ x, ˙ p ℓ ) = D ℓ G (( ˙ x, p ) , ˙ p ℓ ) and D ℓ D j G (( ˙ x, p ) , ˙ x j , ˙ p ℓ ) = 0 for j ≥ x, p, ˙ p ) → D ℓ G (( x, p ) , ˙ p ℓ ) can be ex-tended to continuous maps Φ ℓ,n : f X n + ℓ × V × P → Y n . Indeed, consider fixed p ∈ V , ˙ p ∈ P , x ∈ f X n + ℓ , and a sequence x k ∈ X m converging to x in the norm of X n + ℓ , k x k − x k X n + ℓ → 0. The derivative D ℓ G (( x k , p ) , ˙ p ℓ ) belongs to Y m − ℓ ֒ → Y n for each x k , and in view of the bound (iii) we get(D.36) k D ℓ G (( x k , p ) , ˙ p ℓ ) − D ℓ G (( x k ′ , p ) , ˙ p ℓ ) k Y n ≤ C k x k − x k ′ k X n + ℓ k ˙ p k ℓ , yielding the existence of the limit Φ ℓ,n ( x, p, ˙ p ) := lim k →∞ D ℓ G (( x k , p ) , ˙ p ℓ ) ∈ Y n .This also gives the continuity of the map x → Φ ℓ,n ( x, p, ˙ p ). Combined with thecontinuity ( p, ˙ p ) → D ℓ G (( x, p ) , ˙ p ℓ ) from the condition (ii), we get the continuity of Φ ℓ,n as stated above.To conclude, we introduce the continuous Ψ ,ℓ,n : X × V × X n + ℓ × P → Y n defined by Ψ ,ℓ,n ( x, p, ˙ x, ˙ p ) = Φ ℓ,n ( p, x, ˙ p ) and Ψ ,ℓ,n : X × V × X n + ℓ × P → Y n defined by Ψ ,ℓ,n ( x, p, ˙ x, ˙ p ) = Φ ℓ,n ( p, ˙ x, ˙ p ). For j ≥ Ψ j,ℓ,n ( x, p, ˙ x, ˙ p ) = 0.The assumptions of Lemma D.25 are thus satisfied, allowing us to concludethat G ∈ e C m ( X × V , X , Y ). (cid:3) D.6. A special case of function G not depending on the parameter p In applications of the chain rule it is convenient to also consider the case ofmaps that do not explicitly depend on the parameter p . We get Lemma D.32 . Suppose that G : U × V → Y and ˜ G : U → Y satisfy (D.37) G ( x, p ) = ˜ G ( x ) ∀ ( x, p ) ∈ U × V . Assume that(1) ˜ G ∈ C m ∗ ( U , Y m ) and(2) for ≤ ℓ ≤ m the map ( x, ˙ x ) D ℓ ˜ G ( x, ˙ x ℓ ) can be extended to a contin-uous map from U × X to Y and for ≤ n ≤ m − the restriction ofthis map to U × X n is continuous as a map with values in Y n .Then G ∈ ˜ C m ( U × V , X , Y ) . Moreover (D.38) k G k e C m ( U×V , X , Y ) ≤ M ′ with (D.39) M ′ = inf (cid:8) M : k D j G ( x, ˙ x ℓ ) k Y n ≤ M k ˙ x k lX n ∀ ( x, ˙ x ) ∈ U × X n ∀ ≤ n ≤ m (cid:9) . Proof. First note that D ℓ G = 0 for ℓ = 0. Let φ l, : U × X → Y denote theextension of D l G to U × X and let φ l,n denote the restriction of φ l, to U × X n .Set(D.40) ψ j, ,n ( x, p, ˙ x, ˙ p ) := φ l,n ( x, ˙ x ) , ψ j,l,n ( x, p, ˙ x, ˙ p ) = 0 if l = 0 . Then the assertion follows from Lemma D.25 (cid:3) 30 D. CHAIN RULES D.7. A map in C \ C ∗ and failure of the inverse functions theorem in C ∗ Proposition D.33 . Let H be an infinite dimensional separable Hilbert space.Then there exists G ∈ C ∗ ( H, H ) ∩ C ∞ ( H \ { } , H ) such that G is not Fr´echetdifferentiable at zero. Moreover the exists a function F ∈ C ∗ ( H, H ) which satisfies DF (0 , ˙ x ) = ˙ x but which is not invertible in any neighbourhood of . Proof. Let ( e k ) k ∈ N be an orthonormal basis of H . We will construct G as aconvergent sum(D.41) G ( x ) = X k ∈ N G k ( x ) e k such that • G k ∈ C ∞ ( H ), • the support supp G k of G k is concentrated near 2 − k e k , • supp G k ∩ supp G l = ∅ for k = l , • the gradients ∇ G k are uniformly bounded and converge weakly, but notstrongly, to 0 as k → ∞ .Specifically G k can be defined as follows. Let P k denote the orthogonal projec-tion of H onto the subspace(D.42) X k := { x ∈ H : ( x, e j ) = 0 ∀ j ≤ k − } . Let(D.43) ϕ ∈ C ∞ c (cid:0) ( − , ) (cid:1) , ≤ ϕ ≤ , ϕ (0) = 1 , (D.44) G k ( x ) = 2 − k ϕ ( k k P k x − e k k ) Y j ≤ k − ϕ (cid:16) j + k ( x, e j ) (cid:17) . For k = 0 the product Q j ≤ k − is replaced by 1. Clearly G k ∈ C ∞ ( H ). Moreover(D.45) supp G k ⊂ K k := n x : | ( x, e j ) | ≤ − k + j if j ≤ k − , | ( x, e k ) − − k | ≤ and | P k +1 x | ≤ − k o . We claim that(D.46) K k ∩ K l = ∅ if k = l .To show this we may assume that k < l . If x ∈ K k ∩ K l then the definition of K k implies that ( x, e k ) ≥ − k while the definition of K l yields | ( x, e k ) | ≤ − k + l .Since both inequalities cannot hold simulateneously we get K k ∩ K l = ∅ . Note alsothat(D.47) x ∈ K k = ⇒ | x | ≤ 18 2 − k + 2516 2 − k + 18 2 − k ≤ − k +1 In particular if x = 0 then the ball B | x | / ( x ) intersects only finitely many ofthe sets K k . Hence the sum G = P k G k e k is a finite sum in B | x | / ( x ) and thusdefines a C ∞ map on that set. Thus(D.48) G ∈ C ∞ ( H \ { } , H ) . Moreover G k (0) = 0 and thus G (0) = 0.We now show that(D.49) the directional derivative D G (0 , ˙ x ) exists and equals 0; and that .7. EXAMPLE AND FAILURE OF IMPLICIT FUNCTION THEOREM 131 (D.50) the map ( x, ˙ x ) D G ( x, ˙ x ) is a continuous map from H × H to H .To prove (D.49) we note that G k ( x ) = 0 if | ( x, e k ) | ≤ and | G k ( x ) | ≤ x ∈ H . Thus(D.51) | G k ( x ) | ≤ | ( x, e k ) | . Since each function G k is in C ∞ ( H ) it suffices to show that for each ˙ x ∈ H (D.52) lim m →∞ lim sup t → t (cid:12)(cid:12)(cid:12) X k ≥ m G k ( t ˙ x ) e k (cid:12)(cid:12)(cid:12) = 0 . Now by (D.51) and orthogonality(D.53) (cid:12)(cid:12)(cid:12) X k ≥ m G k ( t ˙ x ) e k (cid:12)(cid:12)(cid:12) = X k ≥ m | G k ( t ˙ x ) | ≤ t X k ≥ m | ( ˙ x, e k ) | = 4 t | P m ˙ x | . Thus(D.54) lim sup t → t (cid:12)(cid:12)(cid:12) X k ≥ m G k ( t ˙ x ) e k (cid:12)(cid:12)(cid:12) ≤ | P m ˙ x | and the assertion (D.52) follows.To prove (D.50) it suffices to prove continuity at (0 , ˙ x ) since we already knowthat G ∈ C ∞ ( H \ { } , H ). Thus we need to show(D.55) lim ( x,v ) → (0 , ˙ x ) D G ( x, v ) = 0 . Since D G is linear in the second argument and since finite linear combinations P Ml =0 a l e l are dense in H it suffices to establish the following two properties(D.56) k D G ( x, v ) k ≤ C k v k ∀ ( x, v ) ∈ H × H, (D.57) lim x → D G ( x, e m ) = 0 ∀ m ∈ N . To prove the bound on D G note that (for x = 0) ∇ G k ( x ) = 2 ϕ ′ ( k k P k x − e k k )(2 k P k x − e k ) Y j ≤ k − ϕ (cid:0) j + k ( x, e j ) (cid:1) (D.58)+ ϕ ( k k P k x − e k k ) X l ≤ k − ϕ ′ (cid:0) l + k ( x, e l ) (cid:1) l − k e l Y j ≤ k − ,j = l ϕ (cid:0) j + k ( x, e j ) (cid:1) . Since the vectors e , . . . , e k − , k P k x − e k are orthogonal this yields, with C ′ =sup | ϕ ′ | ,(D.59) |∇ G k ( x ) | ≤ C ′ 14 + C ′ X l ≤ k − l − k ≤ C ′ . Since the G k have disjoint support and since D G (0 , v ) = 0 it follows that(D.60) k D G ( x, v ) k ≤ √ | ϕ ′ | k v k ∀ ( x, v ) ∈ H × H and thus (D.56).To prove (D.57) note that G k ( x ) = 0 if k x k ≤ − k . Thus for k x k ≤ − m wehave(D.61) | D G ( x, e m ) | ( ≤ m − k if x ∈ supp G k for some k ,= 0 else. 32 D. CHAIN RULES Now if x ∈ supp G k and x → k → ∞ . This implies (D.57).Thus we have shown that(D.62) G ∈ C ∗ ( H, H ) with D G (0 , ˙ x ) = 0 ∀ ˙ x ∈ H. We finally show that G is not Fr´echet differentiable at 0. If G was Fr´echet differ-entiable at 0 the Fr´echet derivative DG (0) would satisfy DG (0) = 0. Thus Fr´echetdifferentiability would give(D.63) lim x → k G ( x ) kk x k = 0 . On the other hand we have(D.64) G (2 − k e k ) = G k (2 − k e k ) e k = 2 − k e k . Taking k → ∞ we get a contradiction to (D.63).To get a counterexample to the inverse function theorem in C ∗ ( H, H ) set(D.65) F ( x ) := x − G ( x ) . Then F ∈ C ∗ ( H, H ) and by (D.62)(D.66) D F (0 , ˙ x ) = ˙ x ∀ ˙ x ∈ H. Now (D.64) imlies that(D.67) F (2 − k e k ) = 0 = F (0)and hence there exists no neighbourhood of 0 in which F is invertible. (cid:3) PPENDIX E Implicit Function Theorem with Loss of Regularity Here we state and prove a version of the implicit function theorem which in-corporates a loss of regularity and is tailored for the use in Chapters 4.5 and 8.We consider a function of three variables (rather than a function of two variablesas in the standard version of the implicit function theorem). The implicit functionwe are looking for expresses the first variable as a function of the second and thethird variable. The reason for this set-up is that the second and the third variableplay very different roles. Differentiation with the respect to the third variable (whichin our application is the renormalised coefficient in the difference operator) leadsto a loss of regularity, while differentiation with respect to the second variable doesnot. This bad behaviour with respect to the third variable is partially compensatedby the fact that we know that F (0 , , p ) = 0 for all values of the third variable in aneighbourhood of 0 (and not just for p = 0) and that we have uniform control of D F (0 , , p ). Theorem E.1 . Let m ≥ . Let X = X m ֒ → . . . ֒ → X , E , and P benormed spaces, with X = ( X m , . . . , X ) , E = ( E , . . . , E ) , and X × E = ( X m × E , . . . , X × E ) . Further, let U ⊂ X , V ⊂ E , and W ⊂ P be open and assumethat F ∈ ˜ C m (( U × V ) × W ; X × E , X ) , i.e., F ∈ C m ∗ ( U × V × W , X ) , for any j ′ + j ′′ + ℓ ≤ m the derivative D j ′ D j ′′ D ℓ F can be extended to a continuous map U × V × W × X j ′ ℓ × E j ′′ × P ℓ → X (E.1) and the restriction of D j ′ D j ′′ D ℓ F defines a continuous map U × V × W × X j ′ n + ℓ × E j ′′ × P ℓ → X n if ≤ n ≤ m − ℓ . (E.2) Assume, moreover, that (0 , , ∈ U × V × W and (E.3) F (0 , , p ) = 0 for all p ∈ W , and, there exists γ ∈ (0 , such that (E.4) k D F (0 , , p ) k L ( X n , X n ) ≤ γ for any n ≤ m and p ∈ W . Then there exist open subsets e U ⊂ U , e V ⊂ V , and f W ⊂ W with ∈ e U , ∈ e V , ∈ f W , and a unique function f : e V × f W → e U such that (E.5) F ( f ( ̟, p ) , ̟, p ) = f ( ̟, p ) for any ( ̟, p ) ∈ e V × f W . Moreover f ∈ ˜ C m ( e V × f W , X ) , i.e., (E.6) f ∈ C n ∗ ( e V × f W , X m − n ) for all ≤ n ≤ m and (E.7) D j ′′ D l f : e V × f W × E j ′′ × P l → X m − l is continuousfor j ′′ + l ≤ m .Finally if F ( x, ̟, p ) = x and ( x, ̟, p ) ∈ e U × e V × f W then x = f ( ̟, p ) . Thederivatives of f are given by the usual formulae, see (E.28) for the first derivativeand the inductive definitions (E.33) and (E.34) for the higher derivatives.If k D j ′ D j ′′ D ℓ F ( x, ̟, p, ˙ x j ′ , ˙ ̟ j ′′ , ˙ p l ) k X n ≤ C k ˙ x k j ′ X n + l k ˙ ̟ k j ′′ E k ˙ p k ℓP . for all ( x, ̟, p ) ∈ U × V × W and all ≤ n ≤ m − ℓ , then there exists a constant C = C ( C , γ, m ) such that (E.8) k D j D ℓ f ( ̟, p, ˙ ̟ j , ˙ p ℓ ) k X m − l ≤ C k ˙ ̟ k j k ˙ p k ℓ for all ( ̟, p ) ∈ e V × f W . The examples in Proposition D.33 shows that the inverse function theorem(and hence the implicit function theorem) in general does not hold in C ∗ , evenwhen there is no loss of regularity. This is why we assume m ≥ Remark E.2 . The usual implicit function theorem also holds in the C m ∗ spacesinstead of the C m spaces as long as m ≥ 2. More specifically, let U ⊂ X , V ⊂ E and assume that F ∈ C m ∗ ( U × V , X ) with F (0 , 0) = 0 and k D F (0 , k ≤ γ < e U ⊂ U and e V ⊂ V and f ∈ C m ∗ ( e V , X ) with f ( e V ) ⊂ e U suchthat F ( f ( ̟ ) , ̟ ) = f ( ̟ ) for all ̟ ∈ e V . This follows directly from Theorem E.1.Indeed, it suffices to consider the situation where X m = . . . = X = X and toextend F trivially to a function on U × V × P which is independent of the thirdargument. Then F satisfies all the hypothesis of Theorem E.1 and the conclusionof the theorem gives the desired assertion. ⋄ Remark E.3 . Let ˆ U = U × V , ˆ X ℓ = X ℓ × E . Then, strictly speaking, thedefinition of e C m (( U × V ) × W , X × E , X ) requires that D j ( x,̟ ) D ℓp F can be extended to a continuous mapˆ U × W × ˆ X j ′ n + l × P ℓ → X n if 0 ≤ n ≤ ℓ − m and j + ℓ ≤ m .(E.9)In view of Corollary D.15 this is equivalent to (E.2). ⋄ Proof. Step 1. Prelimary estimates.We claim that there exist subsets e U ⊂ U , e V ⊂ V , f W ⊂ W that are balls around0 and a constant M such that the following estimates hold:(E.10) k D j ′ D j ′′ D ℓ F (( x, ̟, p ) , ˙ x j ′ , ˙ ̟ j ′′ , ˙ p ) k X n + ℓ ≤ M k ˙ x k j ′ X n k ˙ ̟ k j ′′ E k ˙ p k ℓ P for all ( x, ̟, p ) ∈ e U × e V × f W , all ˙ x ∈ X , ˙ ̟ ∈ E , ˙ p ∈ P , and all j ′ + j ′′ + ℓ = 2,0 ≤ n + ℓ ≤ m ,(E.11) k D F (( x, ̟, p ) , ˙ ̟ ) k X m ≤ M k ˙ ̟ k E for all ( x, ̟, p ) ∈ e U × e V × f W , (E.12) k F (0 , ̟, p ) k X m ≤ M k ̟ k E for all ( ̟, p ) ∈ e V × f W , and . IMPLICIT FUNCTION THEOREM WITH LOSS OF REGULARITY 135 (E.13) k D F ( x, ̟, p ) k L ( X n , X n ) ≤ γ for all ( x, ̟, p ) ∈ e U × e V× f W , ≤ n ≤ m. Indeed, using the joint continuity in (E.2) at ( x, ̟, p ) = 0 and ( ˙ x, ˙ ̟, ˙ p ) = 0 we seethat for ε = 1 there exists a δ ∈ (0 , 1] such that k D j ′ D j ′′ D ℓ F (( x, ̟, p ) , ˙ x j ′ , ˙ ̟ j ′′ , ˙ p ) k X n + ℓ < k ˙ x k X n , k ˙ ̟ k E , k ˙ p k P ) < δ and max( k x k X , k ̟ k E , k p k P ) < δ . By the mul-tilinearity of D j ′ D j ′′ D ℓ this implies (E.10) if M ≥ δ − . Similarly we see that(E.11) holds. Now (E.12) follows from (E.11), the assumption F (0 , , p ) = 0 andLemma D.6. Finally (E.13) follows from the assumption k D F (0 , , p ) k L ( X n , X n ) ≤ γ and (E.10) (applied with ℓ = 0) provided that the radius of e U and e V is chosensufficiently small. Step 2. Existence, uniqueness and continuity of f .First, observe that, according to (E.2), the derivative D F defines a continuousmap D F : e U × e V × f W × X m → X m . Taking into account the inequality (E.13)and, possibly, shrinking the diameters of balls e U , e V , and f W , we have(E.14) k F ( x , ̟, p ) − F ( x , ̟, p ) k X m ≤ γ k x − x k X m for any x , x ∈ e U and any ̟ ∈ e V and p ∈ f W . Employing now the Banach fixedpoint theorem [ Die60 , (10.1.1)] (and possibly shrinking e V and f W further) we getthe existence of a unique map f : e V × f W → e U such that F ( f ( ̟, p ) , ̟, p ) = f ( ̟, p )for any ( ̟, p ) ∈ e V × f W ; moreover, f ∈ C ( e V × f W , X m ). Step 3. Differentiability of f , i.e., f ∈ C ∗ ( e V × f W , X m − ).Using the characterisation in terms of Peano derivatives, Proposition D.7, weneed to find a continuous function f (1) : ( e V × f W ) × ( E × P ) → X m − so that, forany ̟ × p ∈ e V × f W and ˙ ̟ × ˙ p ∈ E × P , we have(E.15) lim t → (cid:13)(cid:13)(cid:13) ξ ( t ) t − f (1) (cid:13)(cid:13)(cid:13) X m − = 0with(E.16) ξ ( t ) := f ( ̟ + t ˙ ̟, p + t ˙ p ) − f ( ̟, p ) . Introducing(E.17) G ( x, ̟, p ) := F ( x, ̟, p ) − x, the function f is defined by(E.18) G ( f ( ̟, p ) , ̟, p ) = 0 for all ( ̟, p ) ∈ e V × f W . Differentiating now formally the equation(E.19) G ( f (( ̟, p ) + t ( ˙ ̟, ˙ p )) , ̟ + t̟, p + t ˙ p )) = 0with respect to t and setting(E.20) R (1)1 := D G (( x, ̟, p ) , ˙ ̟ ) + D G (( x, ̟, p ) , ˙ p )we expect that(E.21) f (1) (( ̟, p ) , ( ˙ ̟, ˙ p )) = − D G ( x, ̟, p ) − R (1)1 with x = f ( ̟, p ). 36 E. IMPLICIT FUNCTION THEOREM WITH LOSS OF REGULARITY The mapping D G ( x, ̟, p ) : X n → X n is bounded and invertible for any n ≤ m since, according to (E.13),(E.22) k D G ( x, ̟, p ) − k L ( X n , X n ) ≤ γ < k D G ( x, ̟, p ) − k L ( X n , X n ) ≤ − γ for any ( x, ̟, p ) ∈ e U × e V × f W . Hence, the function f (1) introduced by (E.21) iswell defined.To verify the claim (E.15), we recall that ξ is continuous (with values in X m )and use the first assertion in Lemma D.27 with l = 1 and Lemma D.6 to estimate(E.24) k G ( x + ξ ( t ) , ̟ + t ˙ ̟, p + t ˙ p ) | {z } =0 − G ( x + ξ ( t ) , ̟ + t ˙ ̟, p ) − D G ( x + ξ ( t ) , ̟ + t ˙ ̟, p, t ˙ p ) k X m − ≤ t sup τ ∈ [0 , k D G ( x + ξ ( t ) , ̟ + t ˙ ̟, p + τ t ˙ p, ˙ p ) − D G ( x + ξ ( t ) , ̟ + t ˙ ̟, p, ˙ p ) k X m − = o ( t ) . Similarly, using the second assertion in Lemma D.27 and Lemma D.6 we get(E.25) k G ( x + ξ ( t ) , ̟ + t ˙ ̟, p ) − G ( x, ̟, p ) | {z } =0 − D G ( x, ̟, p, ξ ( t )) − D G ( x, ̟, p, t ˙ ̟ ) k X m − = o ( t ) + o ( k ξ ( t ) k X m − ) . Combining these two estimate we deduce that(E.26) k D G ( x, ̟, p ) ξ ( t ) + tR (1)1 k X m − ≤ o ( t ) + o ( k ξ ( t ) k X m − ). and using (E.23) and the definition of f (1) it follows that(E.27) k ξ ( t ) − tf (1) k X m − = o ( t ) + o ( k ξ ( t ) k X m − ) . This implies first that k ξ ( t ) k X m − ≤ Ct for small | t | and then division by t yieldsthe desired assertion (E.15).We finally show that(E.28) f (1) (( ̟, p ) , ( ˙ ̟, ˙ p )) = − D G ( x, ̟, p ) − ( D G (( x, ̟, p ) , ˙ ̟ ) + D G (( x, ̟, p ) , ˙ p ))defines a continuos map from e V × f W × × E × P to X m − . Together with (E.15)this show that f ∈ C ∗ ( e V × f W ; X m − ). Clearly the map(E.29) ( ̟, p ) , ( ˙ ̟, ˙ p )) D G (( x, ̟, p ) , ˙ ̟ ) + D G (( x, ̟, p ) , ˙ p )has the desired continuity properties.It thus suffices to verify the following continuity property of D G − for any n with 0 ≤ n ≤ m :(E.30) Whenever ( x j , ̟ j , p j , y j ) → ( x, ̟, p, y ) in e U × e V × f W × X n then D G ( x j , ̟ j , p j ) − y j → D G ( x, ̟, p ) − y in X n . This would be obvious if were able to assume that ( x, ̟, p ) → D G ( x, ̟, p ) iscontinuous as a map with values in L ( X n , X n ). However, we only have continuity . IMPLICIT FUNCTION THEOREM WITH LOSS OF REGULARITY 137 of ( x, ̟, p, ˙ x ) → D G (( x, ̟, p ) , ˙ x ) as a map from e U × e V × f W × X n to X n . Toshow that (E.30) holds under this weaker assumption let z := D G ( x, ̟, p ) − y and z j := D G ( x j , ̟ j , p j ) − y j . Then(E.31) D G (( x j , ̟ j , p j ) , z j − z ) = ( y j − y ) − ( D G (( x j , ̟ j , p j ) , z ) − y ) → X n . Since k D G ( x j , ̟ j , p j ) − k L ( X n , X n ) ≤ / (1 − γ ) it follows that z j → z in X n . Step 4. Higher Peano derivatives and proof of (E.6).Let 2 ≤ k ≤ m . Employing Proposition D.7 again, we will prove that f ∈ C k ∗ ( e V × f W , X m − k ) by showing that f : e V × f W → X m − k has continuous Peanoderivatives up to order k . As before ( ̟, p ) ∈ e V × f W and for sufficiently small t let ξ ( t ) := f ( s + t ˙ ̟, p + t ˙ p ) − f ( ̟, p ). We will show by induction that ξ ( t ) is Peanodifferentiable at 0 and that the Peano derivatives up to order k can be computedby expanding the identity(E.32) 0 = G ( x + ξ ( t ) , ̟ + t ˙ ̟, p + t ˙ p )) , where x = f ( ̟, p ),to order k in t .Define f (1) by (E.21). For k ≥ R k = R k ( t ) = R k ( t, ̟, p, ˙ ̟, ˙ p )and f ( k ) = f ( k ) ( ̟, p, ˙ ̟, ˙ p ) as follows,(E.33) R k ( t ) := X j ′ + j ′′ + ℓ ≤ kj ′′ + ℓ ≥ j ′ ! j ′′ ! ℓ ! D j ′ D j ′′ D ℓ G ( x, ̟, p ) , k − ℓ − j ′′ X q =1 f ( q ) q ! t q ! j ′ , ˙ ̟ j ′′ , ˙ p ℓ ) t j ′′ + ℓ ++ X ≤ j ′ ≤ k j ′ ! D j ′ G ( x, ̟, p ) , (cid:16) k − X q =1 f ( q ) q ! t q (cid:17) j ′ ! . Note that R k is a polynomial in t . We use R ( j ) k to denote its j -th order derivativeat t = 0, i.e., R ( j ) k /j ! is the coefficient of t j in the polynomial R k . Also, notice thatin the right hand side of the equation above, only terms f ( q ) of the order q ≤ k − R k ( t ) contains all the terms of order t j with j ≤ k of thejoint Taylor expansion of G and ξ ( t ) except for the term D G ( x, ̟, p, ξ ( t )). Thuslooking on the coefficients of t k it is natural to define(E.34) f ( k ) := − D G ( x, ̟, p ) − R ( k ) k , i.e., f ( k ) is the unique solution of the linear equation D G ( x, ̟, p, ˙ x ) + R ( k ) k = 0(we will see below that R ( k ) k ∈ X m − k and that this equation has indeed a uniquesolution in X m − k ).For k ≤ m , we will prove by induction that(E.35) f ( k ) ∈ X m − k and that f ( k ) is the sought Peano derivative since(E.36) (cid:13)(cid:13)(cid:13) ξ ( t ) − k X q =1 f ( q ) q ! t q (cid:13)(cid:13)(cid:13) X m − k = o ( t k ) . For k = 1 the definitions of R (1)1 and f (1) agree with those given in Step 3. Theclaims (E.35) and (E.36) for k = 1 were also established in Step 3. 38 E. IMPLICIT FUNCTION THEOREM WITH LOSS OF REGULARITY Assume now that (E.35) and (E.36) hold for k − k ≤ m . Then it iseasy to see that for all t we have R k ( t ) ∈ X m − k and in particular R ( k ) k ∈ X m − k .Indeed, if ℓ + j ′′ ≥ P k − ℓ − j ′′ q =1 f ( q ) q ! t q ∈ X m − k + ℓ and, since(E.37) D j ′ D j ′′ D ℓ G maps U × V × W × X j ′ m − k + ℓ × E j ′′ × P ℓ to X m − k , the first sum in the definition of R k ( t ) is in X m − k . If ℓ = j ′′ = 0, then P k − q =1 f ( q ) q ! t q ∈ X m − k +1 which is mapped by D j ′ G ( x, ̟, p ) into X m − k +1 implying that the secondsum in the definition of R k ( t ) is contained in X m − k +1 ⊂ X m − k . We have seenin Step 3 that the map ˙ x D G (( x, ̟, p ) , ˙ x ) is bounded and invertible as a mapfrom X n to X n for all 0 ≤ n ≤ m . Hence, the definition (E.34) implies that f ( k ) is well defined and lies in X m − k .To prove (E.36), we first define(E.38) e R k ( t ) := X j ′ + j ′′ + ℓ ≤ kj ′′ + ℓ ≥ j ′ ! j ′′ ! ℓ ! D j ′ D j ′′ D ℓ (( x, ̟, p ) , ξ ( t ) j ′ , ˙ ̟ j ′′ , ˙ p ℓ ) t j ′′ + ℓ ++ X ≤ j ′ ≤ k j ′ ! D j ′ G (( x, ̟, p ) , ξ ( t ) j ′ ) . Similar to the estimate for the first derivative, it follows from Lemma D.27, Lemma D.6and Proposition D.7 (c.f. also Lemma D.13) that(E.39) (cid:13)(cid:13)(cid:13) G ( x + ξ ( t ) , ̟ + t ˙ ̟, p + t ˙ p ) | {z } =0 − G ( x, ̟, p ) | {z } =0 − D G (( x, ̟, p ) , ξ ( t )) − e R k ( t ) (cid:13)(cid:13)(cid:13) X m − k ≤≤ sup τ ∈ [0 , (cid:13)(cid:13)(cid:13) X j ′′ + ℓ = k j ′′ ! ℓ ! (cid:16) D j ′′ D ℓ G (( x + τ ξ ( t ) , ̟ + τ t ˙ ̟, p + τ t ˙ p ) , ˙ ̟ j ′′ , ˙ p ℓ ) −− D j ′′ D ℓ G (( x, ̟, p ) , ˙ ̟ j ′′ , ˙ p ℓ ) (cid:17)(cid:13)(cid:13)(cid:13) X m − k t k + sup τ ∈ [0 , (cid:13)(cid:13)(cid:13) X j ′ + j ′′ + ℓ = kj ′≥ j ′ ! j ′′ ! ℓ ! (cid:16) D j ′ D j ′′ D ℓ G (( x, + τ ξ ( t ) , ̟ + τ t ˙ ̟, p + τ t ˙ p ) , ( ξ ( t ) t ) j ′ , ˙ ̟ j ′′ , ˙ p ℓ ) −− D j ′ D j ′′ D ℓ G (( x, ̟, p ) , ( ξ ( t ) t ) j ′ , ˙ ̟ j ′′ , ˙ p ℓ ) (cid:17)(cid:13)(cid:13)(cid:13) X m − k t k The first term on the right hand side is o ( t k ) since D j ′′ D ℓ G is continuous in all ofits arguments and since ξ ( t ) → X m . For the second term we use that ℓ ≤ k − j ′ ≥ ξ ( t ) /t converges to f (1) in X m − . As a result, observing that D j ′ D j ′′ D ℓ is a continuous map from U × V × W × X j ′ m − × E j ′′ × P ℓ to X m − − ℓ ֒ → X m − k , the second term is also o ( t k ). In summary,(E.40) k D G (( x, ̟, p ) , ξ ( t )) + e R k ( t ) k X m − k = o ( t k ) . Combining the induction assumption,(E.41) (cid:13)(cid:13)(cid:13) ξ ( t ) − k − j ′′ − ℓ X q =1 f ( q ) q ! t q (cid:13)(cid:13)(cid:13) X m − k + ℓ + j ′′ = o ( t k − j ′′ − ℓ ) . IMPLICIT FUNCTION THEOREM WITH LOSS OF REGULARITY 139 valid for any j ′′ + ℓ ≥ k P k − j ′′ − ℓq =1 f ( q ) q ! t q k X m − k + ℓ + j ′′ ≤ Ct which follows from (E.41) and the bound k ξ ( t ) k X m − ≤ Ct proven in Step 3, wecan evaluate every term occurring in R k − e R k . Namely, we bound(E.42) (cid:13)(cid:13)(cid:13) D j ′ + j ′ D j ′′ D ℓ G (( x, ̟, p ) , (cid:16) k − ℓ − j ′′ X q =1 f ( q ) q ! t q − ξ ( t ) (cid:17) j ′ , ξ ( t ) j ′ , ˙ ̟ j ′′ , ˙ p ℓ ) t j ′′ + ℓ (cid:13)(cid:13)(cid:13) X m − k = o ( t k ) . Here we took into account that the difference R k − e R k contains only terms with j ′ ≥ o (( t k − j ′′ − ℓ ) j ′ ) t j ′′ + ℓ t j ′ = o ( t k ) since ( k − j ′′ − ℓ ) j ′ + j ′′ + ℓ + j ′ ≥ k + ( j ′ − k − j ′′ − ℓ ) + j ′ ≥ k . Similarly for the remaining terms,(E.43) (cid:13)(cid:13)(cid:13) D j ′ + j ′ G (( x, ̟, p ) , (cid:16) k − X q =1 f ( q ) q ! t q − ξ ( t ) (cid:17) j ′ , ξ ( t ) j ′ ) (cid:13)(cid:13)(cid:13) X m − k = o ( t k )since j ′ ≥ j ′ + j ′ ≥ o (( t k − ) j ′ ) t j ′ = o ( t k ) t ( k − j ′ − j ′ − = o ( t k ).As a result, we can conclude that(E.44) k R k ( t ) − e R k ( t ) k X m − k = o ( t k )and thus(E.45) k D G (( x, ̟, p ) , ξ ( t )) + R k ( t ) k X m − k = o ( t k ) . Moreover one can easily check that for any q ≤ k (E.46) k R q ( t ) − R k ( t ) k X m − k = o ( t q )and thus the derivatives of order q at 0 satisfy R ( q ) q = R ( q ) k . Now the definition of f ( q ) for q ≤ k implies that(E.47) D G (( x, ̟, p ) , f ( q ) ) = − R ( q ) q = − R ( q ) k . Thus(E.48) k D G (( x, ̟, p ) , k X q =1 f ( q ) q ! t q ) + R k ( t ) k X m − k = o ( t k )since R k is a polynomial with values in X m − k . Comparison with (E.45) yields(E.49) k D G (( x, ̟, p ) , ξ ( t ) − k X q =1 f ( q ) q ! t q ) k X m − k = o ( t k )and this implies the claim (E.36) since ˙ x G (( x, ̟, p ) , ˙ x ) is a bounded and invert-ible map from X m − k to itself.We have thus shown that for any n ≤ m the map f : V × W → X m − n hasPeano derivatives for any k ≤ n given by(E.50) f ( k ) (( ̟, p ) , ( ˙ ̟, ˙ p )) = f ( k ) , where f ( k ) is inductively defined by (E.33) and (E.34) with x = f ( ̟, p ). It followsby induction that the maps ( ̟, p, ( ˙ ̟, ˙ p )) R ( k ) k , (E.51) ( ̟, p, ( ˙ ̟, ˙ p )) f ( k ) (E.52)are continuous as maps from e V × f W × E × P to X m − n (here we use again (E.30)). 40 E. IMPLICIT FUNCTION THEOREM WITH LOSS OF REGULARITY Thus f ( n ) exists and is continuous on ( e V × f W , X m − n ). By Proposition D.7,the existence and continuity of Peano derivatives f ( n ) thus finally implies that f ∈ C n ∗ ( e V × f W , X m − n ) for all n ≤ m . Step 5. Improved estimates for D j D ℓ f and proof of (E.7).For j = 0 there is nothing to show since D l f ( ̟, p, ˙ p ℓ ) = f ( l ) ( ̟, p, , ˙ p ) and thus(E.7) follows from (E.6). For j ≥ n := j + ℓ and note that(E.53) 1 n ! f ( n ) ( ̟, p, ˙ ̟, s ˙ p ) = n X l =0 s l j ! 1 ℓ ! D j D l f ( ̟, p, ˙ ̟ j , ˙ p ℓ )Thus, up to a constant factor, D j D ℓ f is given by the coefficient of s l in thepolynomial s f ( n ) ( ̟, p, ˙ ̟, s ˙ p ). Using this observation we will now prove (E.7)by induction over n .For n = 1 the assertion follows directly from (E.28).Assume the assertion has been shown for j + l ≤ n − n ≤ m ). We willshow the assertion for j + l = n . In view of (E.34) it suffices to show the following:If R ( n ) n,l ( ̟, p, ˙ ̟, ˙ p ) is the coefficient of s l in the polynomial h ( s ) := R ( n ) n ( ̟, p, ˙ ̟, s ˙ p )then R ( n ) n,l : e V × f W × E × P → X m − l is continuous.To see this note that h ( s ) is a weighted sum of terms of the form D j ′ D j ′′ D ℓ ′ F ( x, ̟, p, f ( q ) , . . . , f ( q j ′ ) , ˙ ̟ j ′′ , ˙ p ℓ ′ ) s ℓ ′ with f ( q i ) = f ( q i ) ( ̟, p, ˙ ̟, s ˙ p ) and terms of the form D j ′ F ( x, ̟, p, f ( q ) , . . . , f ( q j ′ ) ) . Using (E.53) we see that R ( n ) n,l is a weighted sum of terms T := D j ′ D j ′′ D ℓ ′ F ( x, ̟, p, D a D ℓ f, . . . , D a j ′ D ℓ j ′ f, ˙ ̟ j ′′ , ˙ p ℓ ′ ) with ℓ i ≤ ℓ − ℓ ′ and of terms T := D j ′ F ( x, ̟, p, D a D ℓ f, . . . , D a j ′ D ℓ j ′ f ) with q i ≤ ℓ where D a i D ℓ i f = D a i D ℓ i f ( ̟, p, ˙ ̟ a i , ˙ p l i ) . Now by induction assumption D a i D ℓ i f : e V × f W × E a i × P l i → X m − ( ℓ − ℓ ′ ) is continuous if ℓ i ≤ ℓ − ℓ ′ . Thus T : e V × f W × E × P → X m − ℓ is continuous.Similarly one shows continuity of T . Step 6. Proof of (E.8).This is proved by induction over n = j + l very similar to Step 5. (cid:3) PPENDIX F Geometry of Course Graining We will use two combinatorial lemmas (Lemma 6.15 and 6.16 from [ Bry09 ])proven by Brydges that are for completeness summarised below. Lemma F.1 . Let X ∈ P c k \ S k . Then (F.1) | X | k ≥ (1 + 2 α ( d )) | X | k +1 with α ( d ) = d )(1+6 d ) . For any X ∈ P k we have (F.2) | X | k ≥ (1 + α ( d )) | X | k +1 − (1 + α ( d ))2 d +1 |C ( X ) | with α ( d ) = d )(1+6 d ) . Lemma F.2 . There exist δ = δ ( d, L ) < such that (F.3) X X ∈P c k \S k X = U δ | X | k ≤ for any k ∈ N and any U ∈ P c k +1 . Proof. For any X contributing to the sum we have | X | k ≥ (1 + 2 α ( d )) | X | k +1 and thus(F.4) X X ∈P c k \S k X = U δ | X | k ≤ L d | U | k +1 δ (1+2 α ( d )) | U | k +1 ≤ δ ≤ − Ld α ( d ) . (cid:3) ibliography [AF05] R.A. Adams and J.J.F. 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Mat., , 5–16 (1940). ist of Symbols A ( q ) = P di,j =1 (cid:0) δ i,j + q i,j (cid:1) ∇ ∗ i ∇ j , page 19 A ( q ) k ( A ( q ) k ˙ H, 0) = D T k (0 , , q )( ˙ H, T k ( · , · , q ) at(0 , A ( q ) k ˙ H )( B ′ , ϕ ) = P B ∈B k ( B ′ ) (cid:2) ˙ H ( B, ϕ )+ P x ∈ B P di,j =1 ˙ d i,j ∇ i ∇ ∗ j C ( q ) k +1 (0) (cid:3) ,page 35 α a parameter in the norm k·k Y r , page 38 α = ( α , . . . , α d ), α i ∈ N , i = 1 , . . . , d , a multiindex, page 10 α ( d ) = d )(1+6 d ) from the bound | X | k ≥ (1 + α ( d )) | X | k +1 − (1 + α ( d ))2 d +1 |C ( X ) | for any X ∈ P k , page 71 α ( d ) = d )(1+6 d ) from the bound | X | k ≥ (1 + α ( d )) | X | k +1 − (1 + α ( d ))2 d +1 |C ( X ) | for any X ∈ P k , page 141 | α | = P di =1 α i (for a multiindex α ), page 10 B δ (0) = { u ∈ R d | | u | < δ } , page 10 B n = [0 , n ] d ∩ Z d , page 99 B r ≤ r r , bound on the number of partitions (for Bruno di Fa`a for-mula), page 57 B x the k -block containing x , page 27 B ∗ = the cube of the side (2 d +1 − L k centered at B , the small setneighbourhood of B , page 25 B ( q ) k ( B ( q ) k ˙ K, 0) = D T k (0 , , q )(0 , ˙ K ), linearisation of T k ( · , · , q ) at(0 , B ( q ) k ˙ K )( B ′ , ϕ ) = − P B ∈B k ( B ′ ) Π P X ∈S k X ⊃ B | X | k (cid:16)R X ˙ K ( X, ϕ + ξ ) µ ( q ) k +1 (d ξ ) (cid:17) , page 35 B k = B k (Λ N ) the set of all k -blocks in Λ N , page 25 B k ( X ) the set of k -blocks in X , page 25 B X N σ -algebra on X N induced by the Borel σ -algebra with respect tothe product topology, page 7 β inverse temperature, page 8 c α ,a coefficients in bounds of derivatives of finite range covariancefunction, page 23 C ( q ) the inverse of the operator A ( q ) , page 19 C ( q ) k finite range covariance operator, page 23 C ( q ) k (0 , C ( q ) k ˙ K ) = D T k (0 , , q )(0 , ˙ K ), linearisation of T k ( · , · q ) at (0 , C ( q ) k ( ˙ K )( U, ϕ ) = P B : B ∗ = U (cid:0) − Π (cid:1) P Y ∈S k Y ⊃ B | Y | (cid:16)R X ˙ K ( Y, ϕ + ξ ) µ ( q ) k +1 (d ξ ) (cid:17) ++ P X ∈P c k \S k X = U R X ˙ K ( X, ϕ + ξ ) µ ( q ) k +1 (d ξ ), page 35 e C m ( U × V ) the class of functions G : U × V → M ′ for which the derivative D j ′ D j ′′ D ℓ G is a continuous map U ×V × M j ′′ × f M j ′ r × ( R d × d sym ) ℓ → M ′ r − ℓ , page 34 C ( q ) k finite range covariance function, page 23 b C ( q ) k ( p ) discrete Fourier transform of the kernel C ( q ) k , page 24 C ( X ) the set of all connected components of X , page 25 c < √ 2, the constant from the bound v ( − m ) + v ( m +1) ≤ c m +1 P mx = − m v ( x ) + c (2 m +1) P mx = − m ∂v ( x ) ,page 103 C = C ( p, M, m ), the constant from the discrete Sobolev estimates,e.g.,max x ∈ B n | f ( x ) | ≤ C n − d P Mk =0 k ( n ∇ ) k f k , page 99 ∂ α = Q di =1 ∂ α i i , page 10 e i unit coordinate vectors in R d , page 7 E the map E : ( M , k·k k, ) → ( M |k , |k·k| k, · ) defined by E ( H )( B, ϕ ) =exp {− H ( B, ϕ ) } , page 54 E the Banach space with the norm k·k ζ , page 10 E k expectation with respect to µ k = µ C ( q ) k , page 24 E N ( ϕ ) = P x ∈ T N P di =1 (cid:0) ∇ i ϕ ( x ) (cid:1) , page 7 E q ( ϕ ) = ( A ( q ) ϕ, ϕ ) = P x ∈ T N P di,j =1 (cid:0) δ i,j + q i,j (cid:1) ∇ i ϕ ( x ) ∇ j ϕ ( x ), page 20 f p ( x ) = L − dN/ e i h p,x i , Fourier basis functions, page 23 F ( X )( ϕ ) = F ( X, ϕ ) for F ∈ M ( P k , X ), page 27 F X ( ϕ ) = Q B ∈B k ( X ) F ( B, ϕ ), page 26 F ( X, ϕ ) = F X ( ϕ ) for F ∈ M ( B k , X ), page 26 F ideal Hamiltonian map, page 38 F N irrelevant term of the solution map, page 38 g k,x ( ϕ ) = h P s =2 L (2 s − k sup y ∈ B ∗ x |∇ s ϕ ( y ) | , page 28 g k : k +1 ,x ( ϕ ) = h P s =2 L (2 s − k +1) sup y ∈ B ∗ x |∇ s ϕ ( y ) | , page 28 G k,x ( ϕ ) = h (cid:0) |∇ ϕ ( x ) | + L k |∇ ϕ ( x ) | + L k |∇ ϕ ( x ) | (cid:1) , page 27 γ uN,β (d ϕ ) = Z N,β ( u ) exp (cid:0) − βH uN ( ϕ ) (cid:1) λ N (d ϕ ), random gradient field with Hamil-tonian H uN (with tilt u ), page 8Γ k, A ( X ) = ( A | X | if X ∈ P c k \ S k X ∈ S k . , page 28 h a parameter in the norms |k·k| k,X or k·k k,X,r (via the weight func-tions G k,x and g k,x ), page 28 H ( B, ϕ ) ideal Hamiltonian of the form H ( B, ϕ ) = λ | B | + ℓ ( ϕ ) + Q ( ϕ ) ,page 26 H k +1 ( B ′ , ϕ ) = P B ∈B k ( B ′ ) Π (cid:16) ( R k +1 H k )( B, ϕ ) − P X ∈S ( k ) X ⊃ B | X | k ( R k +1 K k )( X, ϕ ) (cid:17) ,page 32 H k = A − k (cid:0) H k +1 − B k K k (cid:1) , page 38 e H k ( B, ϕ ) = Π (cid:16) ( R k +1 H k )( B, ϕ ) − P X ∈S ( k ) X ⊃ B | X | k ( R k +1 K k )( X, ϕ ) (cid:17) , page 32 H N ( ϕ ) = E N ( ϕ ) + P x ∈ T N P di =1 V ( ∇ i ϕ ( x )), Hamiltonian on T N (withno tilt), page 7 ist of Symbols 147 H uN ( ϕ ) = E N ( ϕ )+ L Nd | u | + P x ∈ T N P di =1 V ( ∇ i ϕ ( x ) − u i ), Hamiltonianon T N with tilt u , page 8 H the initial Hamiltonian map in Theorem 4.9, page 39 H ( x, ϕ ) initial ideal Hamiltonian, page 37 η a parameter in the norm k·k Y r , page 38 η = ( η i ) ∈ R d , page 67 η i,j coefficients of a quadratic test function ˙ ϕ ( x ) = P di,j =1 η i,j ( x − x ) i ( x − x ) j , page 67 η ( d ) = η (2 ⌊ d +22 ⌋ + 8 , d ), page 48 η ( n, d ) = max( ( d + n − , d + n + 6) + 10, the decay exponent in finiterange decomposition, page 23 θ a contractivity constant for operator C ( q ) k , page 36 χ ( X, U ) = ( |{ B ∈B k ( X ): B ∗ = U }|| X | if X ∈ S k (Λ N ) , U = X if X ∈ P k (Λ N ) \ S k (Λ N ) , for any connected U ∈P k +1 , page 31 I k ( B, ϕ ) = exp (cid:8) − H k ( B, ϕ ) (cid:9) , page 30˜ I k ( B, ϕ ) = exp (cid:8) − e H k ( B, ϕ ) (cid:9) , page 30˜ J k ( B, ϕ ) = 1 − ˜ I ( B, ϕ ), page 30 K k +1 ( U, ϕ ) = P X ∈P k ( U ) χ ( X, U ) exp (cid:8) − P B ∈B k ( U \ X ) e H k ( B, ϕ ) (cid:9) R X e K k ( X, ϕ, ξ ) µ k +1 (d ξ ),page 32 K k +1 = S k ( H k , K k , q ) = C k K k + S k ( H k , K k , q ) − D S k ((0 , , q ) , K k ) ,page 38 K κ,p,u ( z ) = Q di =1 (cid:2) p + (1 − p ) exp (cid:8) (1 − κ ) (cid:0) z i − u i ) (cid:9)(cid:3) − 1, Mayer functionfor the potential from [ BK07 ], page 11 K ( q ) ( X, ϕ ) = exp n P x ∈ X P di,j =1 q i,j ∇ i ϕ ( x ) ∇ j ϕ ( x ) o K ( X, ϕ ), page 20 K u ( X, ϕ ) = Q x ∈ X K u ( ∇ ϕ ( x )) with a function K u : R d → R , page 9 K V,β,u ( z ) = exp (cid:8) − β P di =1 U (cid:0) z i √ β , u i (cid:1)(cid:9) − 1, the Mayer function for pertur-bation V , page 8 K V,β,u ( X, ϕ ) = Q x ∈ X K V,β,u ( ∇ ϕ ( x )), page 9 e K k ( X, ϕ, ξ ) = P Y ∈P k ( X ) ( I k ( ϕ + ξ ) − ˜ I k ( ϕ )) X \ Y ( ϕ, ξ ) K k ( Y, ϕ + ξ ), page 30 κ parameter in K κ,p,u , page 11 κ ( d ) = (cid:0) d + η (2 ⌊ d +22 ⌋ + 8 , d ) (cid:1) , page 43 L linear size of a renormalization block, page 7 ℓ ( ϕ ) = P x ∈ B (cid:2)P di =1 a i ∇ i ϕ ( x ) + P di,j =1 c i,j ∇ i ∇ j ϕ ( x ) (cid:3) , linear term ofideal Hamiltonian, page 26 λ N ( L Nd − X N , page 7Λ N = { x ∈ Z d : | x | ∞ ≤ ( L N − } (identified with torus T N ) , page 7 M ( B k , X ) the set of all L k -periodic maps F : B k × X → R such that F ( B, · ) ∈ M ( X , ν k +1 ) for all B ∈ B k , page 26 M ∗ ( B k , X ) the set of all L k -periodic maps F : B k × X → R such that F ( B, · ) ∈ M ( X , ν k +1 ) for all B ∈ B k living on ( B ∗ ) ∗ , page 26 c M r { K ∈ M ( P k , X ) , k K k ( A , B ) k,r < ∞} , page 54 c M : ,r { K ∈ M ( P k , X ) , k K k ( A , B ) k : k +1 ,r < ∞} , page 54 M ( P k , X ) the set of all L k -periodic maps F : P k × X → R such that F ( X, · ) ∈ M ( X , ν k +1 ) for all X ∈ P k , page 25 48 List of Symbols M ( S k , X ) the set of all L k -periodic maps F : S k × X → R such that F ( X, · ) ∈ M ( X , ν k +1 ) for all X ∈ S k , page 26 M ( X N ) set of all functions on X N measurable with respect to λ N , page 24 M ( B k , X ) the set of all ideal Hamiltonians: quadratic functions of the form H ( B, ϕ ) = λ | B | + ℓ ( ϕ ) + Q ( ϕ ) , page 26 M ( X N ) = M ( X N , B X N ), the set of probability measures on X N , page 7 µ ( q ) (d ϕ ) = Z ( q ) N exp (cid:8) −E q ( ϕ ) (cid:9) λ N (d ϕ ), page 19 µ ( q ) k (d ϕ ) Gaussian measure with covariance C ( q ) k , page 20 M = M k, = ( M ( B k , X ) , k·k k, ), page 34 M r = M k,r = ( M r ( P c k , X ) , k·k ( A ) k,r ), page 34 N the power yielding the size (of the torus) L N , page 7 ν (d ϕ ) = ν β =1 (d ϕ ), page 9 ν β (d ϕ ) = Z (0) N,β exp (cid:0) − β E N ( ϕ ) (cid:1) λ N (d ϕ ), Gaussian measure on X N , page 8 ν ( q ) k the measure on X N with covariance C ( q ) k + · · · + C ( q ) N +1 , page 24 ∇ i ϕ ( x ) = ϕ ( x + e i ) − ϕ ( x ), discrete derivative, page 7 ∇ ∗ i ϕ ( x ) = ϕ ( x − e i ) − ϕ ( x ), dual of discrete derivative ∇ i , page 7 |∇ s ϕ ( x ) | = P | α | = s |∇ α ϕ ( x ) | , page 27 p = ( p , . . . , p d ) ∈ b T N , dual variables, page 23 p parameter in K κ, p ,u (replacing β ), page 11 p t = p t ( κ ), corresponding phase transition value, page 12 P P ( e I, e J, e P )( U, ϕ ) = P X ,X ∈P ( U ) X ∩ X ∅ χ ( X ∪ X , U ) e I U \ ( X ∪ X ) ( ϕ ) e J X ( ϕ ) e P ( X , ϕ )mapping ( M ( B k , X ) , |k·k| k ) × ( M ( B k , X ) , |k·k| k ) × ( M ( P c k , X ) , k·k ( A/ k : k +1 ,r )into ( M (( P k +1 ) c , X ) , k·k ( A ) k +1 ,r ) , page 54 P P ( I, K ) = ( I − ◦ K mapping , page 54 P ( P K )( X, ϕ ) = Q Y ∈C ( X ) K ( Y, ϕ ) , page 54 π i the co-ordinate projection π i ( x ) = x i for x ∈ Z d , page 67Π the projection from M ∗ ( B , X ) to M ( B , X ): Π F ( B, ˙ ϕ ) = F ( B, ℓ ( ˙ ϕ ) + Q ( ˙ ϕ, ˙ ϕ ): ℓ agrees with DF ( B, 0) on all quadratic functions˙ ϕ on ( B ∗ ) ∗ and Q agrees with D F ( B, 0) on all affine functions˙ ϕ on ( B ∗ ) ∗ , page 29 P k = P k (Λ N ) the set of all k -polymers in Λ N , page 25 P k ( X ) the set of all polymers Y consisting of subsets of blocks from B k ( X ), page 25 P c k the set of all connected k -polymers, page 25 q a symmetric d × d -matrix, page 19 q ( K u ) the value of q yielding H N = 0, page 21 k q k operator norm of q viewed as operator on R d equipped with ℓ metric, page 23 Q ( ϕ, ϕ ) = P x ∈ B P di,j =1 d i,j ( ∇ i ϕ )( x )( ∇ j ϕ )( x ), quadratic term of idealHamiltonian, page 26 r a bound on the order of derivatives used in the norm k·k ζ , page 10 R R ( P, q )( X, ϕ ) = ( R ( q ) P )( X, ϕ ) = R X P ( X, ϕ + ξ ) µ ( q ) k +1 (d ξ ) map-ping ( M ( P c k , X ) , k·k ( A ) k,r ) × ( R d × d sym , k·k ) into ( M ( P c k , X ) , k·k ( A ) k : k +1 ,r ), page 54 ist of Symbols 149 R R ( H, K, q )( B, ϕ ) = Π (cid:16) ( R ( q ) H )( B, ϕ ) − P X ∈S X ⊃ B | X | ( R ( q ) K )( X, ϕ ) (cid:17) mapping ( M ( B k , X ) , k·k k, ) × ( M ( P c k , X ) , k·k ( A ) k,r ) × ( R d × d sym , k·k )into ( M ( B k , X ) , k·k k, ) , page 54 R k renormalisation maps ( R k F )( ϕ ) = R X N F ( ϕ + ξ ) µ ( q ) k (d ξ ), page 24 ρ ( x, y ) = inf {| x − y + k | ∞ : k ∈ ( L N Z ) d } , page 7 R d × d sym the set of symmetric d × d -matrices, page 19 S k the map S : M ( B k , X ) × M ( P c k , X ) × R d × d sym → M (( P k +1 ) c , X )given by S ( H k , K k , q ) = K k +1 , page 33 S the map S is composed as S ( H, K, q ) = P (cid:0) E ( R ( H, K, q )) , − E ( R ( H, K, q )) , R ( P ( E ( H ) , K ) , q ) (cid:1) , page 54 σ β ( u ) = − lim N →∞ βL dN log Z N,β ( u ), free energy (surface tension) withtilt u , page 8 ς ( u ) = − lim N →∞ L dN log Z N ( u ), the perturbative component of thesurface tension, page 9 ς N ( u ) = − L dN log Z N ( u ), the finite volume perturbative component ofthe surface tension, page 19 S k = S k (Λ N ) = { X ∈ P c k : | X | k ≤ d } , the set of small polymers, page 25 T Taylor expansion up to the second order, T F ( B, ˙ ϕ ) = F ( B, 0) + DF ( B, ϕ ) + D F ( B, ϕ, ˙ ϕ ), page 29 T k map from M ( B k , X ) × M ( P k , X ) × R d × d sym to M ( B k +1 , X ) × M ( P k +1 , X ), T k (( H k , K k )) = ( H k +1 , K k +1 ), page 32 T the map from Y × E × M to Y , page 38 T N = (cid:0) Z /L N Z (cid:1) d , torus, page 7 b T N = (cid:8) p = ( p , . . . , p d ) : p i ∈ {− ( L N − πL N , − ( L N − πL N . . . , , . . . , ( L N − πL N } (cid:9) ,dual torus, page 23 τ a a translation by a vector a ∈ Z d , page 25 u = ( u . . . , u d ) ∈ R d a tilt, page 7 U ( s, t ) = V ( s − t ) − V ( − t ) − V ′ ( − t ) s , page 8 U ρ = { ( H, K ) ∈ M ( B k , X ) × M ( P k , X ) : k H k k, < ρ, k K k ( A ) k,r < ρ } ,page 34 V : R → R potential perturbation, page 7 V = { q ∈ R d × d sym : k q k < / } , page 34 V N = { ϕ : Z d → R ; ϕ ( x + k ) = ϕ ( x ) ∀ k ∈ ( L N Z ) d } , set of fieldstaken as ℓ ( R L Nd ), page 7 w Xk ( ϕ ) = exp nP x ∈ X ω (cid:0) d g k,x ( ϕ ) + G k,x ( ϕ ) (cid:1) + L k P x ∈ ∂X G k,x ( ϕ ) o , theweak weight function, page 28 w Xk : k +1 ( ϕ ) = exp nP x ∈ X (cid:0) (2 d ω − g k : k +1 ,x ( ϕ )+ ωG k,x ( ϕ ) (cid:1) +3 L j P x ∈ ∂X G k,x ( ϕ ) o ,the weak weight function, page 28 W Xk ( ϕ ) = exp (cid:8)P x ∈ X G k,x ( ϕ ) (cid:9) the strong weight function, page 27 ω a parameter in the weightfunction w Xk ( ϕ ), page 28 ∂X = { y X |∃ z ∈ X such that | y − z | = 1 }∪{ y ∈ X |∃ z X such that | y − z | = 1 } , the boundary of X , page 27 50 List of Symbols X the closure of X : the smallest polymer Y ∈ P k +1 of the nextgeneration such that X ⊂ Y , page 25 X ∗ = ∪{ B ∗ : B ∈ B k ( X ) } , the small set neighbourhood of X , page 25 | X | k = |B k ( X ) | , page 25 X N = { ϕ ∈ V N : P x ∈ T N ϕ ( x ) = 0 } , page 7 ξ k a random field distributed according to µ k = µ C ( q ) k , page 24 y an element y = ( H , H , K , . . . , H N − , K N − , K N ) of Y , page 38 y = T ( y , K, H ) the 2( N + 1)-tuple defined by H k and K k , page 38 Z (0) N = Z (0) N,β =1 , page 9 Z (0) N,β = R X N exp (cid:0) − β E N ( ϕ ) (cid:1) λ N (d ϕ ), page 8 Z N,β ( u ) = R X N exp (cid:0) − βH uN ( ϕ ) (cid:1) λ N (d ϕ ), partition function on T N with tilt u , page 8 Z ( q ) N = R X N exp (cid:8) −E q ( ϕ ) (cid:9) λ N (d ϕ ), page 20 ζ a parameter in the exponential weight of a norm (e.g. k·k ζ ) ,page 10 Z N ( u ) = R X N P X K u ( X, ϕ ) ν (d ϕ ), page 9 ◦ ( F ◦ F )( X, ϕ ) = P Y ⊂ X F ( Y, ϕ ) F ( X \ Y, ϕ ), the circle productof F , F ∈ M ( P k , X ), page 26( · , · ) the scalar product ( ϕ, ψ ) = P x ∈ T N ϕ ( x ) ψ ( x ), page 7 | x | ∞ max i =1 ,...,d | x i | , page 7 | x | = pP x i , the Euclidean norm, page 7 |k·k| k |k F k| k = |k F ( B ) k| k,B for F ∈ M ( B k , X ), page 27 |k·k| k,X the weighted strong norm, |k F ( X ) k| k,X = sup ϕ | F ( X, ϕ ) | k,X,r W − Xk ( ϕ ),page 27 k·k k, k H k k, = L dk | λ | + L dk h P di =1 | a i | + L ( d − k h P di,j =1 | c i,j | + h P di,j =1 | d i,j | ,page 29 k·k ( A ) k,r k F k k,r = sup X ∈P c k k F ( X ) k k,X,r Γ k,A ( X ) , r = 1 , . . . , r , , page 28 k·k ( A ) k : k +1 ,r k F k k : k +1 ,r = sup X ∈P c k k F ( X ) k k : k +1 ,X,r Γ k, A ( X ) , r = 1 , . . . , r ,page 28 k·k (b) k,r k F k (b) k,r = k F ( B ) k k,B,r for F ∈ M ( B k , X ), page 28 | · | j,X | S | j,X = sup | ˙ ϕ | j,X ≤ (cid:12)(cid:12) S k ( ˙ ϕ, . . . , ˙ ϕ ) (cid:12)(cid:12) , j = k, k + 1 , for s -linear function S k on X × · · · × X , page 27 | · | j,X,r | F | j,X,r = P rs =0 1 s ! | D s F ( ϕ ) | j,X , j = k, k + 1, for F ∈ C r ( X ),page 27 k·k k,X,r k F ( X ) k k,X,r = sup ϕ | F ( X, ϕ ) | k,X,r w − Xk ( ϕ ) , r = 1 , . . . , r , page 28 k·k k : k +1 ,X,r k F ( X ) k k : k +1 ,X,r = sup ϕ | F ( X, ϕ ) | k,X,r w − Xk : k +1 ( ϕ ) , r = 1 , . . . , r ,page 28 | · | k,X a norm on X : | ϕ | k,X = max ≤ s ≤ sup x ∈ X ∗ h L k (cid:0) d − 22 + s (cid:1)(cid:12)(cid:12) ∇ s ϕ ( x ) (cid:12)(cid:12) ,page 27 | · | k +1 ,X a norm on X : | ϕ | k +1 ,X = max ≤ s ≤ sup x ∈ X ∗ h L ( k +1) (cid:0) d − 22 + s (cid:1)(cid:12)(cid:12) ∇ s ϕ ( x ) (cid:12)(cid:12) ,page 27 k L k = sup {k L ( f ) k : k f k ≤ } , norm of a linear operator L betweenBanach spaces, page 35 ist of Symbols 151 k·k Y r the norm on Y r , k y k Y r = max k ∈{ ,...,N − } η k k H k k k, ∨ max k ∈{ ,...,N } αη k k K k,r k k ,page 38 k·k ζ kKk ζ = sup z ∈ R d P | α |≤ r ζ | α | (cid:12)(cid:12) ∂ α z K ( z ) (cid:12)(cid:12) e − ζ − | z | , norm in the Ba-nach space E , page 10 ϕ (cid:12)(cid:12) X ∗ the restriction of ϕ to X ∗∗