On A Finite Range Decomposition of the Resolvent of a Fractional Power of the Laplacian II. The Torus
aa r X i v : . [ m a t h - ph ] M a y ON A FINITE RANGE DECOMPOSITION OF THE RESOLVENTOF A FRACTIONAL POWER OF THE LAPLACIANII. THE TORUSP. K. Mitter
Laboratoire Charles CoulombCNRS-Universit´e Montpellier- UMR5221Place E. Bataillon, Case 070, 34095 Montpellier Cedex 05 Francee-mail: [email protected]
Abstract : In previous papers, [1], [2], we proved the existence as well as regularity of afinite range decomposition for the resolvent G α ( x − y, m ) = (( − ∆) α + m ) − ( x − y ), for0 < α < m , in the lattice Z d for dimension d ≥
2. In this paper, which is acontinuation of the previous one, we extend those results by proving the existence as wellas regularity of a finite range decomposition for the same resolvent but now on the latticetorus Z d /L N +1 Z d for d ≥ m = 0 and 0 < α <
2. We also prove differentiabilityand uniform continuity properties with respect to the resolvent parameter m . Here L isany odd positive integer and N ≥
1. Introduction
In previous papers [1] and [2] we proved the existence as well as regularity of a finiterange decomposition for the resolvent G α ( x − y, m ) = (( − ∆) α + m ) − ( x − y ) (1 . < α < m , in the lattice Z d for dimension d ≥
2. The definitionand properties of a finite range decomposition were given in [1, 2]. The reference [2]incorporates the content of the published version [1] together with its erratum. and willthus be convenient to refer to. The main result is Theorem 1.1 of [1], restated in [2]. Inthis paper, we will prove for all α in the interval 0 < α < Z d /L N +1 Z d . Thisis the content of Theorem 1.1 below. Continuity and differentiability properties in m aregiven in Theorem 1.2 below for α in the interval 1 < α <
2. We emphasise that Theorems [1] m = 0. Results for finite range decompositions of generalfamilies of massless models on the discrete torus are given in [9].The resolvent (1.1) arises as the covariance of the Gaussian measure underlying variousstatistical/field theoretic systems with long range interactions (see [2]). For d = 2 , d c = 2 α . Thus for α in the above intervalwe can arrange for the system to be below the upper critical dimension, and this is wherenon-trivial critical phenomena for long range systems are expected.For α = 2 the resolvent in (1.1) is that of a standard massive Laplacian. A finite rangedecomposition on the lattice Z d was obtained in [3]. In this case also the methods ofthis paper can be applied for obtaining a finite range decomposition on the lattice torusstarting from the work in [3]. A finite range decomposition for the resolvent of a massiveLaplacian on the lattice torus was obtained earlier in [10] using the results of [4].This paper is a companion to the earlier papers [1, 2]. We will use freely the notationsand results, especially Theorem 1.1, Corollary 1.2 and Proposition 2.1, of these references.However, for the convenience of the reader, before embarking on the proofs of Theorems1.1 and Theorem 1.2 we will give some indications of the strategy to be followed makinguse of results from earlier references.A survey of earlier results and references together with motivation was given in [1]and we will not repeat them here. We simply remind the reader that one of the mainapplications of finite range decompositions is in rigorous Renormalisation Group analysisof statistical/field theoretic systems near and at the critical point of second order phasetransitions. The lattice acts as an ultraviolet cutoff but we also need a finite volume cutoffand then later take the infinite volume limit. In the finite volume theory it is desirable topreserve translation invariance. One convenient way of doing that is putting the theory ona torus of finite period which is the edge length of the fundamental domain in the shape ofa square, cube or hypercube. The goal of extending the finite range decomposition of theresolvent (1.1) given in [1, 2] (together with regularity properties) to the torus is achievedin this paper.We should point out that a different way of achieving the same goal has been given in [6]using estimates from [5]. This is an essential ingredient in [6] where critical exponents belowthe critical dimension d c = 2 α have been studied for the n -component ϕ model with longrange interactions in the regime where ε = d c − d = 2 α − d > [2] Definitions
Let L = 3 p , p ≥ ε j = L − j , j ≥
0, 0 < α < d ≥
2. Let N ≥ T N +1 (= T dN +1 ) denote the torus Z d /L N +1 Z d of edge length L N +1 .The fundamental cube Q N +1 = [ − L N +1 , L N +1 ] d ∩ Z d has the property that every point of Z d has a unique translate with respect to Q N +1 . The volume of the fundamental cube is | Q N +1 | = L ( N +1) d . Functions on the torus are periodic functions. Integration (summation)on the torus is defined as usual as integration over the fundamental cube. Moreover wedefine L ( T N +1 ) = L ( Q N +1 ). If X ⊂ Z d then L ( X ) is the space of summable functionson X .In the following we often speak of periodizing a function. Let f : Z d → R . We say that f has a periodization f T N +1 with period L N +1 with N any positive integer if for ∀ x ∈ Z d the sum f T N +1 ( x ) = X y ∈ L N +1 Z d f ( x + y ) (1 . f ∈ L ( Z d ) then the sum converges absolutely in L ( Q N +1 ) and defines f T N +1 as a function in L ( T N +1 ). For analogous onsiderations in the continuum see e.g. Steinand Weisz, Chapter 7, in [7].Finally we note that we shall often employ continuum integral notations for lattice sums.The Lebesgue measure in ( ε n Z ) d is the counting meaure times ε dn .All objects in the following Theorem 1.1 will be defined and introduced below immediatelyafter the statement of the theorem. Theorem 1.1
Let 0 < α < d ≥ N ≥
2. Let ε j = L − j , ∀ j ≥ m = 0. Then the positivedefinite function G α ( x − y, m ) on Z d has a periodized version G α,T N +1 ( x − y, m ) whichis a function in L ( T N +1 ). Moreover for all m = 0 we have the the following finite rangedecomposition: G α,T N +1 ( x − y, m ) = N − X j =0 L − j [ ϕ ] Γ j,α ( x − yL j , L jα m ) + L − N [ ϕ ] G N,α,T N +1 ( x − yL N , L Nα m )(1 . [3] ϕ ] = d − α . j,α ( · , m ), defined on ( ε j Z ) d , which appear in the sumare those in Theorem 1.1 of [1] and [2]. The function G N,α,T N +1 which did not appear in[1], 2] will be defined later at the end of this theorem. The functions Γ j,α ( · , m ) have finiterange L and satisfy the bounds stated in [1] and [2]:For all j ≥ ≤ q ≤ j , and all p ≥ || ∂ pε j Γ j,α ( · , m ) || L ∞ (( ε q Z ) d ) ≤ c L,p,α (1 + m ) − . (1 . j = 0 , ≤ q ≤ j we have the bound || ∂ pε j Γ j,α ( · , m ) || L ∞ (( ε q Z ) d ) ≤ c L,p,α (1 + m ) − . (1 . ∂ ε j = ∂ ε j ,e k , k = 1 , .., d is a forward lattice partial derivative with increment ε j and in any particular direction e k in the lattice ( ε j Z ) d . Moreover ∂ pε j is a multi-derivativeof order p defined as in the continuum but now with lattice forward derivatives. e , ...., e d are unit vectors which give the orientation of R d as well as the orientation of all embeddedlattices ( ε j Z ) d ⊂ R d . By construction the lattices are nested in an obvious way. Theconstant c L,p,α depends on
L, p, α . It depends implicitly on the dimension d .The functions on Z d ˜Γ j,α ( x, m ) = L − j [ ϕ ] Γ j,α ( xL j , L jα m ) (1 . L j +1 ˜Γ j,α ( x, m ) = 0 : | x | ≥ L j +1 (1 . ≤ j ≤ N − T N +1 . Their periodization give back thefunctions themselves. They satisfy the regularity bounds of Corollary 1.2 of [1] and [2]:for j ≥ || ∂ p Z d ˜Γ j,α ( · , m ) || L ∞ ( Z d ) ≤ c L,p,α (1 + L jα m ) − L − (2 j [ ϕ ]+ pj ) (1 . j = 0 , || ∂ p Z d ˜Γ j,α ( · , m ) || L ∞ ( Z d ) ≤ c L,p,α (1 + L jα m ) − L − (2 j [ ϕ ]+ pj ) . (1 . N ≥ G N,α,T N +1 ( x, m ) = L − N [ ϕ ] G N,α,T N +1 ( xL N , L Nα m ) (1 . [4] L ( T N +1 ) and satisfies for all m = 0 | ∂ p Z d ˜ G N,α,T N +1 ( x, m ) | ≤ c L,α,p L − Nα m − L − (2 N [ ϕ ]+ pN ) . (1 . m ≥ √ C L − Nα where C is any positive constant independent of N , and all integers p ≥
0, we therefore get the bound || ∂ p Z d ˜ G N,α,T N +1 ( · , m ) || L ∞ ( T N +1 ) ≤ c L,p,α L − (2 N [ ϕ ]+ pN ) (1 . c L,p,α depends on
C, L, p, α but is independent of N , m . A guide to Theorem 1.1
We recall for the benefit of the reader the basic objects introduced above. The functionsΓ j,α ( · , m ) : ( ε j Z ) d → R are defined by Γ j,α ( · , m ) = Z ∞ ds ρ α ( s, m ) Γ j ( · , s ) . (1 . j ( · , s ) is the rescaled fluctuation covariance in the finite range decomposition ofthe resolvent of the standard Laplacian (see [3]]) and ρ α ( s, m ) is the spectral functiongiven by Proposition 2.1 in [1, 2]: ρ α ( s, m ) = sin πα/ π s α/ s α + m + 2 m s α/ cos πα/ . (1 . j ( · , s ) of [3] were usedin [1, 2] to provide the bounds on the fluctuation covariances ˜Γ j,α ( · , m ) in Theorem 1.1above.The function ˜ G N,α,T N +1 ( · , m ) is the periodization of a function ˜ G N,α ( · , m ) on Z d whichis shown to be in L ( Z d ). The latter function is the unrescaled version of the function G N,α ( · , m ) on ( ε N Z ) d given by G N,α ( · , m ) = Z ∞ ds ρ α ( s, m ) G N ( · , s ) . (1 . Z d was given for anarbitrary but finite number of terms together with an explicit formula for the remainderwhich is G N ( · , s ). This formula for the remainder is given and used later in Section 2 in the [5] G N,α,T N +1 ( · , m ) in Theorem 1.1 above. Rescalingswere performed in [3] which is why G N ( · , s ) is a function on the lattice ( ε N Z ) d . Remark 1: Scale independence of constants
As in [1] (erratum) and [3], one can get rid of the scale dependence of constants by coarsegraining on a larger scale L ′ = L r with r a large positive integer and holding L fixed. Thefinite range expansion can be rewritten by summing the fluctuation covariances and theremainder over the intermediate scales. The fluctuation covariances on the coarser scale L ′ are defined by ˜Γ ′ j,α ( · , m ) = r − X l =0 ˜Γ l + jr,α ( · , m ) . (1 . G α ( · , m ) = X j ≥ ˜Γ ′ j,α ( · , m ) (1 . ′ j,α ( x − y, m ) = 0 , | x − y | ≥ ( L ′ ) j +1 . (1 . L ′ . Thisis explained in [1] as well as in [2] in the paragraph on coarse graining which followsCorollary 1.2 and is proved in Appendix A of [2]. Remark 2 :The function ˜ G N,α,T N +1 on the torus, introduced earlier, can also be viewed as the sumof all the functions ˜Γ j,α for j ≥ N . In [6] the function was estimated as the the sumof estimates of the summands. Instead we estimate this function directly using its ex-plicit representation together with Fourier analysis and estimates on the discrete Fouriertransform. Remark 3 : The bounds (1.9) and (1.10) on the fluctuation covariances differ from thosegiven in Proposition 10.1 of [7] as was noted earlier in [2] and [3]. In particular the(1 + L jα m ) − term in the bounds occur only for j = 0 , j ≥ j . Howeverthe bound (1.12) agrees with the relevant bound in Proposition 10.1 of [6] once one takesaccount of the scale dimension [ ϕ ] of the Gaussian field ϕ (which is 2[ ϕ ] = d − α ).We have the following continuity and differentiability properties in m of the functionsappearing in the finite range decomposition (1.3) of Theorem 1.1. They are given in [6] α restricted to the interval 1 < α < d ≥ Theorem 1.2 Differentiability of fluctuation covariances : Let 1 < α < d ≥
2. For all m > j ≥
1, the functions ˜Γ j,α ( · , m ) are differentiable functions of m and the derivativessatisfy the bounds:For 1 < α < p ≥ || ∂∂m ∂ p Z d ˜Γ j,α ( · , m ) || L ∞ ( Z d ) ≤ c L,α,p L − pj L − j ( d − ( m ) − − α ) . (1 . Uniform Continuity : As a consequence for all m >
0, ˜Γ j,α ( · , m ) is a uniformly continuousfunction of m . For all m i > i = 1 , < α < p ≥ || ∂ p Z d ˜Γ j,α ( · , m ) − ∂ p Z d ˜Γ j,α ( · , m ) || L ∞ ( Z d ) ≤ c L,α,p L − j ( d − L − pj (cid:12)(cid:12) ( m ) ( − αα ) − ( m ) ( − αα ) (cid:12)(cid:12) . (1 . c L,α,p in (1.22) and (1.21) are independent of j, m , m .2. Differentiability of ˜ G N,α,T N +1 ( · , m ) : For all m = 0 and all integers p ≥ || ∂∂m ∂ p Z d ˜ G N,α,T N +1 ( · , m ) || L ∞ ( Z d ) ≤ c L,α,p L − pN L − ( N +1) d ( m ) − (1 . c L,α,p is independent of N . As a consequence we have the following Uniform continuity of ˜ G N,α,T N +1 ( · , m )For all m i = 0 , i = 1 , p ≥ || ∂ p Z d ˜ G N,α,T N +1 ( · , m ) − ∂ p Z d ˜ G N,α,T N +1 ( · , m ) || L ∞ ( Z d ) ≤ c L,α,p L − pN L − ( N +1) d m − m − | m − m | . (1 . Remark 4: Scale independence of constants in mass derivative estimates :For d ≥ ∀ p ≥ d = 2 , ∀ p ≥ L ′ as in Remark 1 above with L ′ = L r with L ≥ r a large positive integer. The mass differentiability boundon the coarse scale fluctation covariance ˜Γ ′ j,α ( · , m ) is then obtained following Appendix [7] ′ with a new constant c ′ L,α,p independent of L ′ . These statementsare proved in Appendix A of the present paper.For d = 2 with p = 0 the bound (1.20) cannot be employed directly and we have toproceed otherwise. We coarse grain the fluctuation covariances in [3] thus producing alog L ′ dependence in bounds as was first done in [4]. This produces ˜Γ ′ j,α ( · , m ) by thesteps in Section 3 of [1, 2]. The bound (1.6) now holds for ˜Γ ′ j,α ( · , m ) with a new constant c ′ L,α,p log L ′ where c ′ L,α,p is independent of L ′ . Remark 5 :Note that the mass derivative bound given in (1.20) agrees after coarse graining (seeRemark 4 above) with that given in Proposition 10.1 of [6] for d = 3 (once one has takenaccount of the definition of ε = 2 α − d which figures in the bounds in [6]). For d = 2 with p ≥ d = 2 with p = 0 we have a logarithmicscale dependence (as in Remark 4 above) but no logarithmic dependence on m . This toois in contrast to the bound in [6]. These bounds for d = 2 are thus stronger than theestimate in Proposition 10.1 of [6].In the next two sections we will give proofs of the above theorems. Before embarkingon the proofs we indicate the strategy. First we note that provided m = 0 the resolvent G α ( x − y, m ) is in L ( Z d ) and therefore periodizable (see e.g. Theorem 2.4 of Stein andWeiss [8], Ch. 7, page 251) and the periodized version exists as an L ( T N +1 ) function onthe torus. Provided m = 0, the function ˜ G N,α,T N +1 ( · , m ) is the periodized version of an L ( Z d ) function ˜ G N,α which we identified earlier. The periodized function is in L ( T N +1 )and thus has a multiple Fourier series. This is obtained by a Poisson summation formulafor the discrete torus. Fourier analysis for finite Abelian groups is discussed in [8]. We willprove that the Fourier coefficients, supplied by the discrete Fourier transform, have rapiddecay which leads not only to the existence but also to very good uniform differentiabilityproperties of the periodic function. This is at the heart of Theorem 1.1. The continuityresults of Theorem 1.2 will turn out to be relatively easy consequences.
2. Proof of Theorem 1.1
The function G α ( x − y, m ) on Z d is pointwise positive. This follows from the fact that itis the resolvent of an α - stable continuous time L´evy walk x ( α ) t ∈ Z d : G α ( x − y, m ) = Z ∞ dt e − m t E x ( x ( α ) t = y ) . (2 . m = 0 [8] | G α ( · , m ) || L ( Z d ) = Z Z d dx | G α ( x, m ) | = Z Z d dx G α ( x, m )= ˆ G α (0 , m )= 1 m < ∞ . (2 . m = 0 the function G α ( · , m ) is in L ( Z d ) as claimed and the series G α,T N +1 ( x, m ) = X n ∈ Z d G α ( x + n L N +1 , m ) (2 . L ( Q N +1 ) and defines a function in L ( T N +1 ) (see [8],Ch.7, Theorem 2.4). Remark : Since G α ( · , m ) is in L ( Z d ) for m = 0 it follows that there exists a δ > | x | → ∞ , G α ( x, m ) ∼ O ( | x | − ( d + δ ) ). In fact a precise estimate shows that δ = α where 0 < α <
2. However this fact will play no role in the rest of this paper.We now proceed to the proof of the finite range decomposition (1.3) and the bounds statedin Theorem 1.1. We only need to prove the existence of the function ˜ G N,α,T N +1 ( x, m ) of(1.11) in L ( T N +1 ) and the bound (1.12). As in the proof of Theorem 1.1 of [1, 2] givenin section 3 of [1, 2], we will start with the finite range decomposition of the resolvent ofthe Laplacian in Z d given in [3]. We will stop after the first N − G ( x − y, s ) = N − X j =0 L − j ( d − Γ j ( (cid:16) x − yL j , L j s (cid:1) + L − N ( d − G N (cid:0) x − yL N , L N s (cid:1) (2 . j and G N are defined through equations (3.28), (3.29) and (3.30) of [3] (see equations(2.11), (2.12) and (2.13) below). The products in these equations are convolution products.We now proceed as in Section 3 of [1], [2]. We insert the above finite range decompositionwith remainder in the Fourier transform of equation (2.2) of Proposition 2.1 of [1] to get(see equations (3.8)- (3.12) of [1, 2]): G α ( x − y, m ) = N − X j =0 L − j [ ϕ ] Γ j,α ( (cid:16) x − yL j , L jα m (cid:1) + L − N [ ϕ ] G N,α (cid:0) x − yL N , L Nα m (cid:1) (2 . L N +1 ) ofthe functions [9] j,α ( x − y, m ) = L − j [ ϕ ] Γ j,α (cid:0) x − yL j , L jα m (cid:1) (2 . ≤ j ≤ N − T N +1 . Moreover Theorem 1.1 and Corollary 1.2 of [1] (corrected in the erratum) and[2] gives the bounds (1.9) and (1.10) on these functions. It therefore remains to study thetorus boundary function˜ G N,α ( x − y ) = L − N [ ϕ ] G N,α (cid:0) x − yL N , L Nα m (cid:1) (2 . G N,α ( · , m ) : ( ε N Z ) d → R (2 . G N,α ( · , m ) = Z ∞ ds ρ α ( s, m ) G N ( · , s ) (2 . ρ a is the spectral function of Proposition 2.1 of [1]. Let us introduce the notation G N ( s )( x − y ) = G N ( x − y, s ) . (2 . Claim : G N ( s )( x − y ) ≥ . Proof
From equations (3.28) and (3.30) of [4] we have G N ( s ) = A N ( s ) G ε N ( s ) A N ( s ) ∗ . (2 . G ε N ( s )( u − v ) is the resolvent of the laplacian on the lattice ( ε N Z ) d and the products in(2.12) are convolution products with (defective) probability measures: G N ( s )( x − y ) = Z ( ε N Z ) d Z ( ε N Z ) d A N ( s )( x, du ) G ε N ( s )( u − v ) A N ( s )( y, dv ) (2 . A N ( s ) is given by a convolution product of averaging operators: A N ( s ) = N Y m =1 A ε j ,m ( L − ( m − )( s ) (2 . s >
0. Now the action of [10] f is given by equation (3.23) of [4]. It is com-posed of a non-negative constraining function and the action of a Poisson kernel measurewhose action is positivity preserving. Therefore the action of each averaging operator ispositivity preserving and hence their convolution product A N ( s ) is positivity preserving.Finally G ε N ( s )( u − v ) being the resolvent of a random walk in ( ε N Z ) d is pointwise positive.Therefore G N ( s )( x − y ) ≥ ||G N ( s ) || L (( ε N Z ) d ) = Z ( ε N Z ) d dx |G N ( x, s ) | = Z ( ε N Z ) d dx G N ( x, s )= ˆ G N (0 , s ) (2 . G N ( p, s ) = | ˆ A N ( p, s ) | s − ˆ∆ ε N ( p ) (2 . p ∈ B ε N = [ − πε N , πε N ] and ˆ∆ ε N ( p ) is the Fourier transform of the ε N -lattice Lapla-cian. From Appendix B of [2] we have for every integer k ≥ N ≥ | ˆ A N ( p, s ) | ≤ c L,k (1 + s ) − ( p + 1) − k . (2 . | ˆ G N ( p, s ) | ≤ c L,k (1 + s ) − ( p + 1) − k ( s − ˆ∆ ε N ( p )) − (2 . c L,k is independent of N . Therefore for s > ||G N ( s ) || L (( ε N Z ) d ) ≤ c L,k (1 + s ) − s . (2 . ρ ( s, m ) in Proposition 2.1,equation (2.4) of [1], [2] 0 ≤ ρ α ( s, m ) ≤ c α s α/ s α + m (2 . m = 0 ||G N,α ( · , m ) || L (( ε N Z ) d ) ≤ c α Z ∞ ds s α/ s α + m ||G N ( · , s ) || L (( ε N Z ) d ) ≤ c α,L,k Z ∞ ds s α/ − s α + m (1 + s ) − ≤ c α,L,k c α m . (2 . [11] c α = Z ∞ ds s α/ − (1 + s ) − < ∞ (2 . < α <
2. It follows from (2.7) and the bound in (2.21) that for m = 0 || ˜ G N,α ( · , m ) || L ( Z d ) = L Nα ||G N,α ( · , L Nα m ) || L (( ε N Z ) d ) ≤ c L,α,k c α L Nα L Nα m . (2 . m = 0, ˜ G N,α ( · , m ) is in L ( Z d ) and˜ G N,α,T N +1 ( x, m ) = X y ∈ L N +1 Z d ˜ G N,α ( x + y, m ) (2 . L ( Q N +1 ) and hence defines a function in L ( T N +1 ).As a L ( T N +1 ) periodic function ˜ G N,α,T N +1 ( · , m ) has a Fourier series which is obtainedby Poisson summation with discrete Fourier transform for the discretized torus:˜ G N,α,T N +1 ( x, m ) = 1 | Q N +1 | X p ∈ πLN +1 Q N +1 ˆ˜ G N,α ( p, m ) e ip.x . (2 . Q N +1 = [ − L N +1 , L N +1 ] d ∩ Z d . Here L = 3 p where p ≥ p is not be confused with the p appearing in the sum). Thus the sum is over adiscretization of the Brillouin zone B ε = [ − π, π ] d where the discrete Fourier transform in Z d occuring in (2.25) is defined. We shall now estimate the Fourier coefficients.From the definition ˜ G N,α, ( x, m ) = L − N [ ϕ ] G N,α ( xL N , L Nα m ) (2 . G N,α ( p, m ) = L Nα ˆ G N,α ( L N p, L Nα m )= L Nα Z ∞ ds ρ α ( s, L Nα m ) ˆ G N ( L N p, s ) (2 . p ∈ [ − π, π ] d and ρ α is that of (1.16). Using the bounds supplied in (2.18) and (2.20)we obtain for m = 0 [12] ˆ˜ G N,α ( p, m ) | ≤ c α c L,k L Nα Z ∞ ds s α/ − s α + L Nα m (1 + s ) − (( L N p ) + 1) − k ≤ c α,L,k L Nα ( L Nα m ) (( L N p ) + 1) − k . (2 . G N,α,T N +1 ( x, m ) = 1 | Q N +1 | X p ∈ Q N +1 ˆ˜ G N,α ( 2 πL N +1 p, m ) e i πLN +1 p.x . (2 . l with respect to x ∈ Z d we get ∂ l Z d ˜ G N,α,T N +1 ( x, m ) | = 1 | Q N +1 | X p ∈ Q N +1 ( 2 πpL N +1 ) l ˆ˜ G N,α ( 2 πL N +1 p, m ) e i πLN +1 p.x (2 . ∂ l Z d is in multi-index notation, p l = Q di =1 p l i i , l i ≥ l = P di =1 l i ≥
0. We now use the bound (2.28) and extend the sumto Z d to get | ∂ l Z d ˜ G N,α,T N +1 ( x, m ) | ≤ c α,L,k L − ( N +1) d L Nα ( L Nα m ) X p ∈ Z d ( 2 π | p | L N +1 ) l ( (2 πpL ) + 1) − k . (2 . k is at our disposal. We choose 2 k > d + l + 1. Then the seriesconverges and we get the bound for all m = 0 and all integers l ≥ | ∂ l Z d ˜ G N,α,T N +1 ( x, m ) | ≤ c L,α,l L − Nα m − L − (2 N [ ϕ ]+ lN ) . (2 .
3. Proof of Theorem 1.2
Throughout the proof we restrict α to the range 1 < α < j,α ( · , m ) : ( ε j Z ) d → R defined in equation (3.11) in Section 3 of [1, 2]. [13] j,α ( · , m ) = Z ∞ ds ρ α ( s, m ) Γ j ( · , s )where Γ j ( · , s ) is the rescaled fluctuation covariance in the finite range decomposition ofthe resolvent of the standard Laplacian (see Section 3 of [1, 3]). Then we have for all0 ≤ j ≤ N −
1, on using the uniform bound in Theorem 5.5 of [4] together with Sobolevembedding: || ∂∂m ∂ pε j Γ j,α ( · , m ) || L ∞ (( ε q Z ) d ) ≤ Z ∞ ds | ∂∂m ρ α ( s, m ) | || ∂ pε j Γ j ( · , s ) || L ∞ (( ε q Z ) d ) ≤ c L,p Z ∞ ds | ∂∂m ρ α ( s, m ) | (1 + s ) − (3 . ρ a ( s, m ) is the spectral function of Proposition 2.1 of [1]: ρ α ( s, m ) = sin πα/ π s α/ s α + m + 2 m s α/ cos πα/ . (3 . | ∂∂m ρ α ( s, m ) | ≤ c α s α/ ( m + s α/ )( s α + m + 2 m s α/ cos πα/ ≤ c α s α/ ( m + s α/ )( s α + m ) (3 . d α ( s, m ) = s α + m + 2 m s α/ cos πα/ ≥ c ′ α ( m + s α ) . Therefore || ∂∂m ∂ pε j Γ j,α ( · , m ) || L ∞ (( ε q Z ) d ) ≤ c L,α,p Z ∞ ds s α/ ( m + s α/ )( s α + m ) (1 + s ) − . (3 . s α/ = m σ we get with a different constant c Lα,p || ∂∂m ∂ pε j Γ j,α ( · , m ) || L ∞ (( ε q Z ) d ) ≤ c L,α,p ( m ) α − H α ( µ ) (3 . µ = ( m ) α (3 . [14] H α ( µ ) = Z ∞ dσ σ α (1 + σ )(1 + σ ) (1 + µσ α ) − . (3 . < α < H α ( µ ) ≤ H α (0) (3 . H α (0) = Z ∞ dσ σ α (1 + σ )(1 + σ ) < ∞ (3 . < α < H α (0) is a constant c α of O (1).From (3.5), (3.6), (3.7), (3.8) and (3.9) we get || ∂∂m ∂ pε j Γ j,α ( · , m ) || L ∞ (( ε q Z ) d ) ≤ c L,α,p ( m ) α − (3 . || ∂∂m ∂ p Z d ˜Γ j,α ( · , m ) || L ∞ ( Z d ) ≤ c L,α,p L − pj L − j [ ϕ ] L jα ( L jα m ) − − α ) = c L,α,p L − j ( d − L − pj ( m ) − − α ) (3 . G N,α,T N +1 ( · , m ).Recall that in the uniform continuity statement m >
0. We will first give an uniform upperbound for its derivative with respect to m from which the uniform Lipshitz continuitybound will follow. To this end we start from the Fourier series representation (2.30) wherethe Fourier coefficients decay rapidly as in (2.28). We shall show presently that theirderivatives with respect to m also decay rapidly. We have || ∂∂m ∂ l Z d ˜ G N,α,T N +1 ( · , m ) || L ∞ ( Z d ) ≤ | Q N +1 | X p ∈ Q N +1 ( 2 πpL N +1 ) l | ∂∂m ˆ˜ G N,α ( 2 πL N +1 p, m ) | . (3 . k ≥ [15] (cid:12) ∂∂m ˆ˜ G N,α ( 2 πL N +1 p, m ) | (cid:12)(cid:12) ≤ c L,k L Nα Z ∞ ds (cid:12)(cid:12) ∂∂m ρ α ( s, m ) (cid:12)(cid:12) m → L Nα m × (1 + s ) − (cid:0) ( 2 πpL ) + 1 (cid:1) − k s − . (3 . (cid:12)(cid:12) ∂∂m ρ α ( s, m ) (cid:12)(cid:12) ≤ c α s α/ ( m + s α/ )( s α + m ) . (3 . Z d we obtain bychoosing k sufficiently large so that the series converges, || ∂∂m ∂ l Z d ˜ G N,α,T N +1 ( · , m ) || L ∞ ( Z d ) ≤ c L,α,l L − ( N +1) d L − Nl L Nα × Z ∞ ds s α/ − ( m + s α/ )( s α + ( m ) ) (1 + s ) − (cid:12)(cid:12) m → L Nα m (3 . F α ( m ) = Z ∞ ds s α/ − ( m + s α/ )( s α + ( m ) ) (1 + s ) − (3 . α > m = 0. We now change variables as in the line before (3.5): s α/ = m σ to get with µ = ( m ) α F α ( m ) = ( m ) − α Z ∞ dσ (1 + σ )(1 + σ ) (1 + µσ /α ) − ≤ ( m ) − α Z ∞ dσ (1 + σ )(1 + σ ) ≤ c α ( m ) − (3 . F α ( L Nα m ) ≤ L − Nα c α ( m ) − . (3 . l ≥ || ∂∂m ∂ l Z d ˜ G N,α,T N +1 ( · , m ) || L ∞ ( Z d ) ≤ c L,α,l L − ( N +1) d L − Nl ( m ) − . (3 . [16] Appendix A
In this Appendix we prove the statements in the first paragraph of Remark 4. By definitionthe fluctuation covariances on the coarser scale L ′ = L r with L ≥ r a largepositive integer is given by (1.17):˜Γ ′ j,α ( · , m ) = r − X l =0 ˜Γ l + jr,α ( · , m ) . (3 . || ∂∂m ∂ p Z d ˜Γ ′ j,α ( · , m ) || L ∞ ( Z d ) ≤ r − X l =0 || ∂∂m ∂ p Z d ˜Γ l + jr,α ( · , m ) || L ∞ ( Z d ) ≤ c L,α,p ( m ) − − α ) r − X l =0 L − p ( l + jr ) L − ( l + jr )( d − ≤ c L,α,p ( m ) − − α ) ( L ′ ) − pj ( L ′ ) ( d − j ∞ X l =0 L − ( p +( d − l . For d ≥ ∀ p ≥ d = 2, ∀ p ≥ ∞ X l =0 L − l = (1 − L ) − and hence || ∂∂m ∂ p Z d ˜Γ ′ j,α ( · , m ) || L ∞ ( Z d ) ≤ c ′ L,α,p ( m ) − − α ) ( L ′ ) − pj ( L ′ ) ( d − j . which is (1.20) with a new constant independent of L ′ as claimed. Acknowledgements
I wish to thank David Brydges for many helpful conversations andfor setting me right on Poisson summation for a discrete torus. I also thank the diligentreviewers for their remarks, questions and suggestions.
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