On paracomposition and change of variables in Paradifferential operators
aa r X i v : . [ m a t h . A P ] F e b ON PARACOMPOSITION AND CHANGE OF VARIABLES INPARADIFFERENTIAL OPERATORS
AYMAN RIMAH SAID
Abstract.
In this paper we revisit the hypothesis needed to define the ”para-composition” operator, an analogue to the classic pull-back operation in the lowregularity setting, first introduced by S. Alinhac in [3]. More precisely we do soin two directions. First we drop the diffeomorphism hypothesis. Secondly wegive estimates in global Sobolev and Zygmund spaces. Thus we fully generalizeBony’s classic paralinearasition Theorem giving sharp estimates for compositionin Sobolev and Zygmund spaces. In order to prove that the new class of operationsbenefits of symbolic calculus properties when composed by a paradifferential oper-ator, we discuss the pull-back of pseudodifferential and paradifferential operatorswhich then become Fourier Integral Operators. In this discussion we show thatthose Fourier Integral Operators obtained by pull-back are pseudodifferential orparadifferential operators if and only if they are pulled-back by a diffeomorphismi.e a change of variable. We give a proof to the change of variables in paradiffer-ential operators.
Keywords—
Composition, Sobolev spaces, Zygmund spaces, Paracom-position, Paradifferential operators, Pseudodifferential operators, Fourierintegral operators, Change of variables.
Contents
1. Introduction 12. Notations and functional analysis 43. Notions of microlocal analysis 74. Pull-back of pseudo and para- differential operators 165. Paracomposition 20References 291.
Introduction
Given a ρ > C ρ diffeomorphism χ : Ω → Ω between two open subsetsof R d , Alinhac constructed an operator χ ∗ : D ′ (Ω ) → D ′ (Ω ) having analogousproperties to the usual composition u → u ◦ χ but with limited dependency on theregularity of χ as for classical paradifferential operators i.e the paraproduct T a iswell defined from H s → H s , for all s for a merely in L ∞ .Alinhac’s construction was motivated by questions that arose from the study of nonlinear PDEs for example: the study of the transport of a distribution’s wave frontby a diffeomorphism with low regularity as in the works of E. Leichtnam in [13], thestudy of the singularities of solutions to semi-linear hyperbolic evolution problemsand the characteristic surfaces of the associated operators(here having low regu-larity), the main reference being Bony’s work on the subject ([6],[7],[8],[9]). Morerecently in [1] and [2], the Paracomposition appears naturally as the ”good vari-able” after a low regularity change of variable in treating the Cauchy problem forthe Water Waves system with rough data. It also appears in our recent proof of thequasi-linearity of the Water Waves system [16]. PhD student at CMLA, Batiment Laplace 61, Avenue du President Wilson 94235 Cachan Cedex.email: [email protected] . The so called good unknown of Alinhac. inally the construction of χ ∗ gives a complete linearization formula to the com-position of two functions(with one being a diffeomorphism) generalizing the classicpara-linearization Theorem by Bony [6] in a low regularity case:Bony showed for u ∈ C ∞ and χ ∈ H sloc , s > d (without the diffeomorphism hypoth-esis): u ◦ χ = T u ′ ( χ ) χ + remainder , and Alinhac showed for u ∈ C σloc , σ > χ ∈ C ρ , ρ > u ◦ χ = χ ∗ u + T u ′ ( χ ) χ + remainder . (1.1)Another fundamental result obtained by Alinhac is that the operator χ ∗ benefitsfrom symbolic calculus properties, that is, it conjugates paradifferential operators.Given T h a paradifferential operator, Alinhac proved a result in the form: χ ∗ T h u = T h ∗ χ ∗ u + remainder , where h ∗ is the pulled back symbol in the case of diffeomorphisms.The main result of this work generalizes Bony’s and Alinhac’s work by: • dropping the diffeomorphism hypothesis with a new operator χ ⋆ : D ′ (Ω ) → D ′ (Ω ). χ ⋆ will coincide with Alinhac’s operator χ ∗ modulo a regular re-mainder in the case of diffeomorphisms. • Giving estimates in ”global” spaces which were of interest for us in our studyof the flow map regularity associated to the Water Waves system.We will then show that χ ⋆ benefits of symbolic calculus properties, for that we willstart by discussing the pull-back of pseudodifferential and paradifferential operatorsby χ which then become Fourier integral operators. In this discussion we showthat those Fourier Integral Operators obtained by pull-back are pseudodifferentialor paradifferential operators if and only if they are pulled-back by a diffeomorphismi.e a change of variable. We also give a proof to the change of variables in paradif-ferential operators as we could not find a reference in the literature.A main application of the paracomposition operator and the paralinearizationformula (1.1) now that we have the global estimates is the study of compositionin Sobolev spaces with limited regularity. The literature on this problem is richand our knowledge of it is certainly incomplete but we mainly looked on two recentarticles treating this subject [5] and [12] in which they study composition in Sobolevspaces and the geometry of diffeomorphisms groups on manifolds. We will limit thediscussion here to the Euclidean space in which the tools presented here significantlyimprove upon the results from [5] and [12]. First in [12] the composition estimatesare proven on H n ( R d ) × D s ( R d ) with n ∈ N , s > d an integer and D s ( R d ) = n ψ − id ∈ H s ( R d ) , ψ is a diffeomorphism o . Here we generalize this to n, s real number and from the paralinearization formula(1.1) it is justified to work in the class D s ( R d ) which appears naturally but it admitsseveral generalization the simplest one is for example using Zygmund spaces, wealso clarify the need of the diffeomorphism hypothesis. More precisely we have thefollowing, Corollary 1.1.
Consider two real numbers s ∈ R , ρ ∈ R ∗ + \ N , and take φ ∈ H s ( R d ) and consider χ ∈ W ρ, ∞ loc ( R d ) a diffeomorphism such that Dχ ∈ W ρ, ∞ ( R d ) . Then φ ◦ χ ∈ H min ( s,ρ ) ( R d ) . The result we have is even stronger indeed it’s a Kato-Ponce like decompositionof the different terms that appear in the H s estimates of composition, for example eeping the notations of the previous Corollary and taking ψ ∈ D s ( R d ) we can haveestimates of the form: k φ ◦ ψ k H s ≤ k Dψ k L ∞ k ψ k H s + k Dφ k L ∞ k ψ − Id k H s . So if only working with Sobolev spaces more sophisticated versions of the previousinequality give,
Corollary 1.2.
Consider a real number s > d , and take φ ∈ H s ( R d ) andconsider χ ∈ W s − d , ∞ loc ( R d ) a diffeomorphism such that Dχ ∈ W , ∞ ( R d ) and D χ ∈ H s − ( R d ) . Then φ ◦ χ ∈ H s ( R d ) . Secondly in [5] to prove the well posedness of EPDIFF equation they treat the caseof change of variables in pseudodifferential operator with a diffeomorphism with lim-ited regularity. The results are restricted to skew-symmetric operators with compactsupport and a diffeomorphism in the class D s ( R d ). Here with the paradifferentialcalculus and the paracomposition in hand, the more general case of symbols withlimited regularity is treated, the pseudodifferential symbols being the the case wherethe symbols are regular, the ellipticity and symmetry hypothesis dropped and theneed of diffeomorphisms justified. More precisely we have Corollary 1.3.
Consider a real number r , A ∈ S r ( R d × R d ) and χ ∈ W s − d , ∞ loc ( R d ) a diffeomorphism such that Dχ ∈ W , ∞ ( R d ) and D χ ∈ H s − ( R d ) . Then the pullback A ∗ of A by χ defined as u ∈ S , A ∗ u = [ A (cid:0) u ◦ χ (cid:1) ] ◦ χ − , is extended to a linear bounded operator from H s ( R d ) to H s − r ( R d ) . Heuristics behind Paradifferential calculus and Paracomposition.
Forthe sake of this discussion let us pretend that ∂ x is left-invertible with a choice of ∂ − x that acts continuously from H s to H s +1 . We follow here analogous ideas to theones presented by Shnirelman in [17]. Paraproduct . One way to define the paraproduct of two functions f, g ∈ H s with s sufficiently large is: we differentiate f g k times, using the Leibniz formula, andthen restore the function f g by the k -th power of ∂ − x : f g = ∂ − kx ∂ kx ( f g )= ∂ − kx (cid:0) g∂ kx f + k∂ x g∂ k − x f + · · · + k∂ x f ∂ k − x g + g∂ kx f (cid:1) = T g f + T f g + R, where, T g f = ∂ − kx (cid:0) g∂ kx f (cid:1) , T f g = ∂ − kx (cid:0) f ∂ kx g (cid:1) , and R is the sum of all remaining terms.The key observation is that if s > + k , then g T f g is a continuous operatorin H s for f ∈ H s − k . The remainder R is a continuous bilinear operator from H s to H s +1 .The operator T f g is called the paraproduct of g and f and can be interpreted asfollows. The term T f g takes into play high frequencies of g compared to those of f and demands more regularity in g ∈ H s than f ∈ H s − k thus the term T f g bearsthe ”singularities” brought on by g in the product f g . Symmetrically T g f bears the”singularities” brought on by f in the product f g and the remainder R is a smootherfunction ( H s +1 ) and does not contribute to the main singularities of the product.Notice that this definition uses a ”general” heuristic from PDE that is the worstterms are the highest order terms (ones involving the highest order of differentiation). aracomposition . We again work with f ∈ H s and g ∈ C s with s large andconsider the composition of two functions f ◦ g which bears the singularities of both f and g , and our goal is to separate them. We proceed as before by differentiating f ◦ g k times, using the Fa´a di Bruno’s formula, and then restore the function f g bythe k -th power of ∂ − x : f ◦ g = ∂ − kx ∂ kx ( f ◦ g )= ∂ − kx (cid:0) ( ∂ kx f ◦ g ) · ( ∂ x g ) k + · · · + ( ∂ x f ◦ g ) · ∂ kx g (cid:1) = g ∗ f + T ∂ x f ◦ g g + R, where, g ∗ f = ∂ − kx (cid:0) ( ∂ kx f ◦ g ) · ( ∂ x g ) k (cid:1) is the paracomposition of f by g and R is the sum of all remaining terms.Again the key observation is that if s > + k , then f g ∗ f is a continuousoperator in H s for g ∈ C s − k . Thus this term bears essentially the singularities of f in f ◦ g . As before T ∂ x f ◦ g g bears essentially the singularities of g in f ◦ g . Theremainder R is a continuous bilinear operator from H s to H s +1 . Thus we haveseparated the singularities of the composition f ◦ g . Change of variable in Paradifferential operators . From what we have seenpreviously it seems likely that the adequate change of variable for paradifferentialoperators is one that comes from commuting with the paracomposition by a diffeo-morphism. We carry on the previous computation with the trivial paradifferentialoperator ∂ x = T iξ and we suppose moreover that g is a diffeomorphism. g ∗ ∂ x f = ∂ − kx (cid:0) ( ∂ k +1 x f ◦ g ) · ( ∂ x g ) k (cid:1) = ∂ − kx (cid:0) ∂ kx [ ∂ − kx ( ∂ k +1 x f ◦ g ) · ( ∂ x g ) k +1 ] · ( ∂ x g ) − (cid:1) = T ( ∂ x g ) − T iξ g ∗ f, and we notice that ( ∂ x g ) − iξ = ( ∂ x ) ∗ is the usual pull-back formula for pseudo-differential symbols by a diffeomorphism g , giving us the desired symbolic calculusrules.1.2. Structure of the paper.
Given the technical nature of the results in this pa-per we start the paper by a quick overview in sections 2 and 3 of notions of functionalanalysis and microlocal analysis . Then in section 4 we present the different resultson the change of variables in pseudodifferential and paradifferential operators. Andfinally with all of the tools needed we redefine the para-composition in section 5 andshow that it satisfies all of the desired properties. Thus the reader interested in thechange of variable/pull-back can directly go to section 4 and if she/he is interestedonly in the para-composition she/he can go to section 5.1.3. Acknowledgement.
I would like to express my sincere gratitude to my thesisadvisor Thomas Alazard.2.
Notations and functional analysis
We present the definitions of the functional spaces that will be used.We will use the usual definitions and standard notations for the regular functions C k , C k for those with compact support, the distribution space D ′ , E ′ for those withcompact support, D ′ k , E ′ k for distributions of order k, Lebesgue spaces ( L p ), Sobolevspaces ( H s , W p,q ) and the Schwartz class S and it’s dual S ′ . All of those spacesare equipped with their standard topologies. Thus the reader familiar with those notions can skip the preliminary sections. efinition 2.1 (Littlewood-Paley decomposition) . Pick P ∈ C ∞ ( R d ) so that P ( ξ ) =1 for | ξ | < and 0 for | ξ | > . We define a dyadic decomposition of unity by:for k ≥ , P ≤ k ( ξ ) = Φ (2 − k ξ ) , P k ( ξ ) = P ≤ k ( ξ ) − P ≤ k − ( ξ ) . Thus, P ≤ k ( ξ ) = X ≤ j ≤ k P j ( ξ ) and ∞ X j =0 P j ( ξ ) . Introduce the operator acting on S ′ ( R d ) : P ≤ k u = F − ( P ≤ k ( ξ ) u ) and u k = F − ( P k ( ξ ) u ) . Thus, u = X k u k . Finally put { k ≥ , C k = supp P k } the set of rings associated to this decomposition. Remark 2.1.
An interesting property of the Littlewood-Paley decomposition is thateven if the decomposed function is merely a distribution the terms of the decomposi-tion are regular, indeed they all have compact spectrum and thus are entire functions.On classical functions spaces this regularization effect can be ”measured” by the fol-lowing inequalities due to Bernstein.
Proposition 2.1 (Bernstein’s inequalities) . Suppose that a ∈ L p ( R d ) has its spec-trum contained in the ball {| ξ | ≤ λ } . Then a ∈ C ∞ and for all α ∈ N d and q ≥ p ,there is C α,p,q (independent if λ ) such that k ∂ αx a k L q ≤ C α,p,q λ | α | + dp − dq k a k L p . In particular, k ∂ αx a k L q ≤ C α λ | α | k a k L p , and for p = 2 , p = ∞k a k L ∞ ≤ Cλ d k a k L . Proposition 2.2.
For all µ > , there is a constant C such that for all λ > and for all α ∈ W µ, ∞ with spectrum contained in {| ξ | ≥ λ } . one has the followingestimate: k a k L ∞ ≤ Cλ − µ k a k W µ, ∞ . Definition 2.2 (Singular support) . f ∈ S ′ ( R d ) is said to be C ∞ in a neighborhoodof x, if there exists a neighborhood ω of x such that for all ψ ∈ C ∞ ( ω ) we have ψf ∈ C ∞ ( R d ) .The singular support of a distribution f, sing supp f , is defined as the complementaryof such points and is clearly closed. Definition 2.3 (Zygmund spaces on R d ) . For r ∈ R we define the space C r ∗ ( R d ) ⊂ S ′ ( R d ) by: C r ∗ ( R d ) = (cid:26) u ∈ S ′ ( R d ) , k u k r = sup q qr k u q k ∞ < ∞ (cid:27) equipped with its canonical topology giving it a Banach space structure.It’s a classical result that for r / ∈ N , C r ∗ ( R d ) = W r, ∞ ( R d ) the classic H¨older spaces.We define the local spaces: C r ∗ ,loc ( R d ) = n u ∈ S ′ ( R d ) , ∀ ψ ∈ C ∞ ( R d ) , ψu ∈ C r ∗ ( R d ) o . roposition 2.3. Let B be a ball with center 0. There exists a constant C suchthat for all r > and for all ( u q ) q ∈ N ∈ S ′ ( R d ) verifying: ∀ q, supp ˆ u q ⊂ q B and (2 qr k u q k ∞ ) q ∈ N is boundedthen , u = X q u q ∈ C r ∗ ( R d ) and k u k r ≤ C − − r sup q ∈ N qr k u q k ∞ . For the definition of spaces in open subsets of R d we follow the presentation of[10]. Let Ω be an open subset of R d . Definition 2.4 (Zygmund spaces on Ω) . For r ∈ R we define the space C r ∗ (Ω) ⊂ D ′ (Ω) by: C r ∗ (Ω) = n u ∈ D ′ (Ω) , u = U | Ω f or some U ∈ C r ∗ ( R d ) o equipped with its canonical topology i.e k u k C r ∗ (Ω) = inf U ∈ C r ∗ ( R d ) U | Ω = u k U k C r ∗ ( R d ) giving it a Banach space structure.We define the local spaces: C r ∗ ,loc (Ω) = (cid:8) u ∈ D ′ (Ω) , ∀ ψ ∈ C ∞ (Ω) , ψu ∈ C r ∗ (Ω) (cid:9) . Definition 2.5 (Sobolev spaces on R d ) . It is also a classical result that for s ∈ R : H s ( R d ) = ( u ∈ S ′ ( R d ) , | u | s = (cid:18) X q qs k u q k L (cid:19) < ∞ ) with the right hand side equipped with its canonical topology giving it a Hilbert spacestructure and | | s is equivalent to the usual norm on k k H s .We define the local spaces: H sloc ( R d ) = n u ∈ S ′ ( R d ) , ∀ ψ ∈ C ∞ ( R d ) , ψu ∈ H s ( R d ) o . Proposition 2.4.
Let B be a ball with center 0. There exists a constant C suchthat for all s > and for all ( u q ) ∈ N ∈ S ′ ( R d ) verifying: ∀ q, supp ˆ u q ⊂ q B and (2 qs k u q k L ) q ∈ N is in L ( N ) then , u = X q u q ∈ H s ( R d ) and | u | s ≤ C − − s (cid:18) X q qs k u q k L (cid:19) . Definition 2.6 (Sobolev spaces on Ω) . For s ∈ R we define the space H s (Ω) ⊂ D ′ (Ω) by: H s (Ω) = n u ∈ D ′ (Ω) , u = U | Ω f or some U ∈ H s ( R d ) o equipped with its canonical topology i.e k u k H s (Ω) = inf U ∈ H s ( R d ) U | Ω = u k U k H s ( R d ) giving it a Hilbert space structure .We define the local spaces: H sloc (Ω) = (cid:8) u ∈ D ′ (Ω) , ∀ ψ ∈ C ∞ (Ω) , ψu ∈ H s (Ω) (cid:9) . This is not immediate from the definition but is a consequence of the fact that H s (Ω) can beseen as a quotient of H s ( R d ) by a closed subset, for a full presentation see [10]. emark 2.2. This definition of the functions in an open subset might not seemas the most natural, in fact there are different ways(intrinsically, extrinsically, byinterpolation etc...) to define H s (Ω) and when no regularity assumption is put on Ω and they don’t necessarily match. In [10] they show that when Ω has Lipschitzregularity all the different definitions of H s (Ω) coincide. We recall the usual nonlinear estimates in Sobolev spaces: • If u j ∈ H s j ( R d ) , j = 1 ,
2, and s + s > u u ∈ H s ( R d ) and if s ≤ s j , j = 1 , s ≤ s + s − d , then k u u k H s ≤ K k u k H s k u k H s , where the last inequality is strict if s or s or − s is equal to d . • For all C ∞ function F vanishing at the origin, if u ∈ H s ( R d ) with s > d then k F ( u ) k H s ≤ C ( k u k H s ) , for some non decreasing function C depending only on F.Finally we present a classic result for operator estimates by Y.Meyer [15]: Lemma 2.1 (Meyer multipliers) . Let δ ∈ R , and suppose we have a sequence m p ∈ C ∞ with, for all k ∈ N , X | α | = k k ∂ α m p k ∞ ≤ C k p ( k + δ ) . The mapping M : u P m p u p = M u maps H s to H s − δ and C r ∗ to C r − δ ∗ for all s, r > δ , with operators norms depending only on the C k for k ≤ E ( s − δ ) + 1 or k ≤ E ( r − δ ) + 1 . Notions of microlocal analysis
In this paragraph we start by reviewing classic notations and results about pseudo-differential calculus, Fourier integral operators and paradifferential calculus, whichcan be found in [11] , [18], [4] and [14] as an accessible presentation to the theoriesand from which we follow the presentation.3.1.
Pseudodifferential Calculus.
We introduce here the basic definitions andsymbolic calculus results. We first introduce the classes of regular symbols.
Definition 3.1.
Given m ∈ R , ≤ ρ ≤ and ≤ σ ≤ we denote the symbol class S mρ,σ ( R d × R d ) the set of all a ∈ C ∞ ( R d × R d ) such that for all multi-orders α, β wehave the estimate: (cid:12)(cid:12)(cid:12) ∂ αx ∂ βξ a ( x, ξ ) (cid:12)(cid:12)(cid:12) ≤ C α,β (1 + | ξ | ) m − ρβ + σα .S mρ,σ ( R d × R d ) is a Fr´echet space with the topology defined by the family of semi-norms: M mβ,α ( a ) = sup R d (cid:12)(cid:12)(cid:12) ∂ αx ∂ βξ a ( x, ξ )(1 + | ξ | ) ρβ − m − σα (cid:12)(cid:12)(cid:12) . Set S m ( R d × R d ) = S m , ( R d × R d ) ,S −∞ ( R d × R d ) = \ m ∈ R S m ( R d × R d ) and S + ∞ ( R d × R d ) = [ m ∈ R S m ( R d × R d ) equipped with their canonically induced topology. iven a symbol a ∈ S m ( R d × R d ), we define the pseudodifferential operator:Op( a ) u ( x ) = a ( x, D ) u ( x ) = (2 π ) − n Z R d e ix.ξ a ( x, ξ )ˆ u ( ξ ) dξ. For u ∈ S ( R d ) we haveOp( a ) u ( x ) = (2 π ) − d Z R d e ix.ξ a ( x, ξ )ˆ u ( ξ ) dξ = (2 π ) − d Z R d e ix.ξ a ( x, ξ ) Z R d e − iy.ξ u ( y ) dydξ = Z R d (cid:18) (2 π ) − n Z R d e i ( x − y ) .ξ a ( x, ξ ) dξ (cid:19) u ( y ) dy. Thus giving us the following Proposition.
Proposition 3.1.
For a ∈ S m ( R d × R d ) , Op( a ) has a kernel K defined by K ( x, y ) = (2 π ) − d Z R d e i ( x − y ) .ξ a ( x, ξ ) dξ = (2 π ) − n F ξ a ( x, y − x ) . (3.1) Which can be inverted to give: a ( x, ξ ) = F y → ξ K ( x, x − y ) = Z R d e − iy.ξ K ( x, x − y ) dy = ( − d e − ix.ξ Z R d e iy.ξ K ( x, y ) dy. (3.2) Definition 3.2.
Let m ∈ R , an operator T is said to be of order m if, and only if,for all µ ∈ R , it is bounded from H µ ( R d ) to H µ − m ( R d ) . Theorem 3.1. If a ∈ S m ( R d × R d ) , then a ( x, D ) is an operator of order m. Moreoverwe have the norm estimate: k a ( x, D ) k H µ → H µ − m ≤ CM mµ,m + d/ ( a ) . We will now present the main results in symbolic calculus associated to pseudo-differential operators.
Theorem 3.2.
Let m, m ′ ∈ R d , a ∈ S m ( R d × R d ) and b ∈ S m ′ ( R d × R d ) . • Composition: Then
Op( a ) ◦ Op( b ) is a pseudodifferential operator of order m + m ′ with symbol a b defined by: a b ( x, ξ ) = (2 π ) − d Z R d × R d e i ( x − y ) . ( ξ − η ) a ( x, η ) b ( y, ξ ) dydη. Moreover,
Op( a ) ◦ Op( b )( x, ξ ) − Op( X | α | Op( a ) , that will note Op( a ) t to avoid con-fusion with the pullback operator defined in this work, is a pseudodifferentialoperator of order m with symbol a t defined by: a t ( x, ξ ) = (2 π ) − d Z R d × R d e − iy.ξ a ( x − y, ξ − η ) dydη Moreover, Op( a t )( x, ξ ) − Op( X | α | We can now write simply: a b ∼ X | α | i | α | α ! ( ∂ αξ a ( x, ξ ))( ∂ αx b ( x, ξ )) and a t ∼ X | α | i | α | α ! ( ∂ αξ ∂ αx ¯ a ( x, ξ )) . Proposition 3.2 (Pseudo-local property) . Let a ∈ S m ( R d × R d ) and let K be itskernel. Then K is C ∞ for x = y . In particular, for all u ∈ S ′ : sing supp a ( x, D ) u ⊂ sing supp u Proof. Let x = y , ψ, θ ∈ C ∞ ( R d ), ψ = 1 near x , θ = 1 near y and supp ψ ∩ supp θ = ∅ . Then ˜ K ( x, y ) = ψ ( x ) K ( x, y ) θ ( y ) is the kernel of the operator ψaθ . By Theorem3.2, ψaθ ∼ −∞ which finishes the proof. (cid:3) Let Ω be an open subset of R d . We will now define the notion of local symbolsand operators in an open set. Definition 3.4 (Local operators and symbols) . We define S m (Ω × R d ) to be the setof a ∈ C ∞ (Ω × R d ) such that for all multi-orders α, β we have the estimate: (cid:12)(cid:12)(cid:12) ∂ αx ∂ βξ a ( x, ξ ) (cid:12)(cid:12)(cid:12) ≤ C α,β (1 + | ξ | ) m − ρβ + σα .S m (Ω d × R d ) is a Fr´echet space with the topology defined by the family of semi-norms: M mβ,α ( a ) = sup Ω × R d (cid:12)(cid:12)(cid:12) ∂ αx ∂ βξ a ( x, ξ )(1 + | ξ | ) ρβ − m − σα (cid:12)(cid:12)(cid:12) . We define the local spaces: S mloc (Ω × R d ) = n a ∈ C ∞ (Ω × R d ) , ∀ ψ ∈ C ∞ (Ω) , ψa ∈ S m (Ω × R d ) o , equipped with its canonical topology giving it a Fr´echet space structure. If a ∈ S m (Ω × R d ) or S mloc (Ω × R d ), the usual formula Au ( x ) = a ( x, D ) u ( x ) = (2 π ) − d Z R d e ix.ξ a ( x, ξ )ˆ u ( ξ ) dξ defines an operator respectively from S ′ ( R d ), E ′ (Ω) to D ′ (Ω), which can be restrictedto an operator E ′ (Ω) → D ′ (Ω) and C ∞ (Ω) → C ∞ (Ω).The link between such operators and the operators obtained by cut-off from globaloperators is given by the following Proposition: Proposition 3.3. Let A : v → C ∞ (Ω) be a continuous linear operator such thatfor all ψ, θ ∈ C ∞ (Ω) , ψAθ ∈ Op( S m ) . Then there exists a ′ ∈ S m (Ω × R d ) withA=a’(x,D)+R, where R is an operator with kernel in C ∞ (Ω × Ω) . roof. Let ( ψ j ) ∈ C ∞ (Ω) be a partition of unity locally finite over Ω. Put ψ j Aψ k = A jk ∈ Op( S m ) then Au = X j,k ψ j Aψ k = X j,k supp ψ j ∩ ψ k = ∅ A jk + X j,k supp ψ j ∩ ψ k = ∅ A jk . Then a ′ = X j,k supp ψ j ∩ ψ k = ∅ A jk ∈ S m (Ω × R d )because for ∀ ψ ∈ C ∞ (Ω) , ψa ′ is a finite sum by definition of a partition of unitylocally finite.The remainder has a kernel: X j,k, supp ψ j ∩ ψ k = ∅ ψ j ( x ) K ( x, y ) ψ k ( y ) ∈ C ∞ (Ω × Ω)by the pseudo-local property, Proposition 3.2. (cid:3) We see from the previous definition that there is subtlety with the support of thefunctions if one want for example to define A t . The following class of local operatorsclarifies that problem: Definition 3.5 (Properly supported operators) . A continuous linear operator A : C ∞ (Ω) → C ∞ (Ω) is said to be properly supported if, for any compact subset K ⊂ Ω ,there exists a compact subset K ′ ⊂ Ω with: supp u ⊂ K = ⇒ supp Au ⊂ K ′ and u = 0 on K ′ = ⇒ Au = 0 on K We see that such an operator maps C ∞ to C ∞ and for example A t can be ex-tended in a standard way to an operator from D ′ (Ω) to itself. Proposition 3.4. Let A = a ( x, D ) where a ∈ S mloc (Ω × R d ) . There exists an operatorR with kernel in C ∞ (Ω × Ω) such that A+R is properly supported.Proof. This is the same proof as Proposition 3.3 because X j,k, supp ψ j ∩ ψ k = ∅ A jk is properly supported. (cid:3) Remark 3.2. The previous Proposition tells us that for local regularity consider-ations we can essentially work with properly supported operators for local symbols(modulo a C ∞ kernel) and by Proposition 3.3 we can do the same for operatorsobtained by cut-off. Fourier Integral Operators. Here we will give basic definitions and resultsas presented in part 1 of H¨ormander’s [11].We wish to define operators of the form : A ω u ( x ) = Z e iS ( x,ξ ) a ( x, ξ )ˆ u ( ξ ) dξ (3.3)= Z e i ( S ( x,ξ ) − y.ξ ) a ( x, ξ ) u ( y ) dy = Z e iω ( x,y,ξ ) a ( x, ξ ) u ( y ) dydξ here u is a regular function, a is a symbol and ω is a given function defining theoperator A . We can clearly see that for example ω = 0 the integral in not definedfor symbols with m ≥ − d , we thus start by the following definition of suitable phasefunctions: Definition 3.6. Let ω ( x, y, ξ ) be a C ∞ (Ω × Ω × R d ) map which is positively homo-geneous of degree one with respect to ξ . Put: R ω = n ( x, y ) ∈ Ω × Ω , ∀ ξ ∈ R d \ { } , ω ( x, y, ξ ) has no critical point o , and its compliment C ω , which is the projection on Ω × Ω of the conic set (with respectto ξ ) of: C = n ( x, y, ξ ) ∈ Ω × Ω × R d \ { } , Dω ξ ( x, y, ξ ) = 0 o . • Then ω is called a phase function on R ω × R d . • ω is called a non-degenerate phase function if at any point in C, the differ-entials D ( ∂ω∂ξ j ) , j = 1 , ..., d, are linearly independent. • ω is called an operator phase function on R ω × R d if for each fixed x (or y)it has no critical point ( y, ξ ) ( or ( x, ξ )) with ξ = 0 . • For U ⊂ Ω define C ω U = { x, ( x, y ) ∈ C ω f or some y ∈ U } . The main example here are pseudodifferential operators with ω ( x, y, ξ ) = ( x − y ) .ξ ,in that case C ω is equal to the diagonal { ( x, x ) , x ∈ Ω } , and we see that all of theprevious definitions naturally apply in this case.The following Proposition will give a definition to the weak form of (3.3): < A ω u, v > = < op ω ( a ) u, v > = Z e iω ( x,y,ξ ) a ( x, y, ξ ) u ( y ) v ( x ) dxdydξ, u, v ∈ C ∞ (Ω) . (3.4) Proposition 3.5. Take a symbol a ∈ S mρ,σ (Ω × Ω × R d ) , ρ > , σ < , and a phasefunction ω on Ω × Ω × R d (i.e R ω = Ω × Ω ). Then:(1) The oscillatory integral (3.4) exists and is a continuous bilinear form for the C k topologies on u,v if m − kρ < − N, m − k (1 − σ ) < − N. Thus we obtain a continuous linear map A from C k (Ω) to D ′ k (Ω) which hasa distribution kernel K ω ∈ D ′ k (Ω × Ω) given by the oscillatory integral K ω ( u ) = Z e iω ( x,y,ξ ) a ( x, y, ξ ) u ( x, y ) dxdydξ, u ∈ C ∞ (Ω × Ω) . (2) If ω has no critical point ( y, ξ ) for each fixed x, then (3.3) is defined as anoscillatory integral and we obtain a continuous map A : C k (Ω) → C (Ω) . Bydifferentiation under the integral sign it follows that A is also continuousmap from C k (Ω) to C j (Ω) if m − kρ < − N − j, m − k (1 − σ ) < − N − j. (3) If ω has no critical point ( x, ξ ) for each fixed y, then the adjoint of Ais defined and has the properties listed in point 2, so A is a continuousmap of E ′ j (Ω) into D ′ k (Ω) . In particular A defines a continuous map from E ′ (Ω) into D ′ (Ω) . R ω is clearly open. 4) The oscillatory integral: K ω ( x, y ) = Z e iω ( x,y,ξ ) a ( x, ξ ) dξ defines a C ∞ (Ω × Ω = R ω ) map , it follows that A is an integral operator with C ∞ kernel, so A is a continuousmap of E ′ (Ω) to C ∞ (Ω) .(5) We have the generalization of the pseudo-local property: sing supp op ω ( a ) u = C ω sing supp u. When ω is an operator phase function it verifies all the previous properties. Proposition 3.6. Let ω ( x, y, ξ ) be a C ∞ (Ω × Ω × R d ) map which is positively homo-geneous of degree one with respect to ξ and a be a symbol in S mρ,σ (Ω × Ω × R d ) , ρ > σ and that either ω is linear or that ρ + σ = 1 . Suppose that a vanishes of infinite orderon C then we have the same results as in the previous Proposition with m replacedby m − ρ + σ .If a just vanishes on C then we can find b ∈ S m − δ + ρρ,σ (Ω × Ω × R d ) such that we havethe formal equality op ω ( a ) u = op ω ( b ) u . As H¨ormander summed up, when ω is non degenerate the singularities of thedistribution u → op ω ( a ) u only depend on the Taylor expansion of a on the set C.The following Proposition, taken from part 2 of [11], gives the natural link betweenpseudodifferential operators and Fourier Integral operators defined by the phasefunction ω ( x, y, ξ ) = ( x − y ) .ξ . Proposition 3.7. Consider a real number m and a symbol c ∈ S m (Ω × Ω × R d ) ,then: a ( x, ξ ) = Z Ω × R d c ( x, y, η ) e i ( x − y ) . ( η − ξ ) dydη ∈ S m (Ω × R d ) and we have: ∀ u ∈ C ∞ (Ω) , op ( x − y ) .ξ ( c ) u = Op( a ) u = (2 π ) − d Z R d e ix.ξ a ( x, ξ )ˆ u ( ξ ) dξ. Moreover the asymptotic expansion of a is given by: ∀ N ∈ N , a ( x, ξ ) − X | α | In the previous setting c is often called an amplitude. We will not give the proof of these Propositions here but we will present thefundamental Lemma behind those results and the idea behind it. The main problemis to define oscillatory integrals of the form: Z e iω ( x,ξ ) a ( x, ξ ) u ( x ) dxdξ, u ∈ C ∞ (Ω) , We start by remarking that the integral is absolutely convergent if a is of order m < − N . Lemma 3.1. If ω has no critical point ( x, ξ ) with ξ = 0 , then one can find a firstorder differential operator L = X j h j ∂ξ j + ˜ h j ∂x j + c with h j ∈ S (Ω × R d ) and ˜ h j , c ∈ S − (Ω × R d ) such that L t e iω = e iω .L is a continuous map from S mρ,σ (Ω × Ω × R d ) to S m − ǫρ,σ (Ω × Ω × R d ) where ǫ = min ( ρ, − σ ) . aking a symbol a of order m we compute: Z e iω ( x,ξ ) a ( x, ξ ) u ( x ) dxdξ = Z e iω ( x,ξ ) La ( x, ξ ) u ( x ) dxdξ = Z e iω ( x,ξ ) L k a ( x, ξ ) u ( x ) dxdξ, under the hypothesis ρ > σ < ǫ > L k a ∈ S m − kǫρ,σ (Ω × Ω × R d ),taking m − kǫ < − N and applying the previous remark we see that the integral isthen well defined.3.3. Paradifferential Calculus. We start by the definition of symbols with limitedspatial regularity. Let W ⊂ S ′ be a Banach space. Definition 3.7. Given ρ ≥ and m ∈ R , Γ m W ( R d ) denotes the space of locallybounded functions a ( x, ξ ) on R d × ( R d \ , which are C ∞ with respect to ξ for ξ = 0 and such that, for all α ∈ N d and for all ξ = 0 , the function x ∂ αξ a ( x, ξ ) belongsto W and there exists a constant C α such that, ∀ | ξ | > , (cid:13)(cid:13) ∂ αξ a ( ., ξ ) (cid:13)(cid:13) W ≤ C α (1 + | ξ | ) m −| α | Given a symbol a , define the paradifferential operator T a by d T a u ( ξ ) = (2 π ) − d Z R d ψ ( ξ − η, η )ˆ a ( ξ − η, η )ˆ u ( η ) dη, where ˆ a ( η, ξ ) = R e − ix.η a ( x, ξ ) dx is the Fourier transform of a with respect to thefirst variable; ψ is a fixed C ∞ function such that:(1) there are ǫ and ǫ such that 0 < ǫ < ǫ < ( ψ ( ξ, η ) = 1 for | ξ | ≤ ǫ (1 + | η | ) ,ψ ( ξ, η ) = 0 for | ξ | ≥ ǫ (1 + | η | ) . (2) for all ( α, β ) ∈ N d × N d , there is C α β such that ∀ ( ξ, η ) : (cid:12)(cid:12)(cid:12) ∂ αξ ∂ βη ψ ( ξ, η ) (cid:12)(cid:12)(cid:12) ≤ C α,β (1 + | ξ | ) −| α |−| β | . Such a ψ is called an admissible cut-off function.For quantitative estimates we introduce as in [14]: Definition 3.8. For m ∈ R , ρ ≥ and a ∈ Γ m W ( R d ) , we set M m W ( a ) = sup | α |≤ d +1+ c sup | ξ |≥ (cid:13)(cid:13)(cid:13) (1 + | ξ | ) m −| α | ∂ αξ a ( ., ξ ) (cid:13)(cid:13)(cid:13) W , where c ≥ is a constant fixed by a choice that generally depends on W . We willessentially work with W = W ρ, ∞ and write Γ m W = Γ mρ and M mW ρ, ∞ ( a ) = M mρ ( a ) with c = ρ . The main features of symbolic calculus for paradifferential operators given by thefollowing Theorems. Theorem 3.3. Take m ∈ R . if a ∈ Γ m ( R d ) , then T a is of order m. Moreover, forall µ ∈ R there exists a constant K such that k T a k H µ → H µ − m ≤ KM m ( a ) . Theorem 3.4. Take m, m ′ ∈ R , and ρ > , a ∈ Γ mρ ( R d ) and b ∈ Γ m ′ ρ ( R d ) . Composition: Then T a T b is a paradifferential operator of order m + m ′ and T a T b − T a b is of order m + m ′ − ρ where a b is defined by: a b = X | α | <ρ i | α | α ! ∂ αξ a∂ αx b Moreover, for all µ ∈ R there exists a constant K such that k T a T b − T a b k H µ → H µ − m − m ′ + ρ ≤ KM mρ ( a ) M m ′ ρ ( b ) . • Adjoint: The adjoint operator of T a , that we will note T ta to again avoidconfusion with the pull back operator defined in this work, is a paradifferentialoperator of order m with symbol a t defined by: a t = X | α | <ρ i | α | α ! ∂ αξ ∂ αx ¯ a Moreover, for all µ ∈ R there exists a constant K such that (cid:13)(cid:13) T ta − T a t (cid:13)(cid:13) H µ → H µ − m + ρ ≤ KM mρ ( a ) . If a = a ( x ) is a function of x only, the paradifferential operator T a is called a para-product. With a good choice of ( ǫ , ǫ ) in the definition of the cut-off function withrespect to our choice of the dyadic decomposition of unity in the Littlewood-Paleydecomposition we get that when a = a ( x ), T a takes the usual form: T a u = ∞ X k =1 Φ k − au k . It follows from Theorem 3.4 and the Sobolev embeddings that: • If a ∈ H α ( R d ) and b ∈ H β ( R d ) with α, β > d , then T a T b − T ab is of order − (cid:18) min { α, β } − d (cid:19) . • If a ∈ H α ( R d ) with α > d , then T ta − T a t is of order − (cid:18) α − d (cid:19) . An important feature of para-products is that they are well defined for function a = a ( x ) which are not L ∞ but merely in some Sobolev spaces H r with r < d . Proposition 3.8. Take m > . If a ∈ H d − m ( R d ) and u ∈ H µ ( R d ) then T a u ∈ H µ − m ( R d ) . Moreover, k T a u k H µ − m ≤ K k a k H d − m k u k H µ A main feature of para-products is the existence of para-linearization Theoremswhich allow us to replace nonlinear expressions by paradifferential expressions, atthe price of error terms which are smoother than the main terms. Theorem 3.5. Let α, β ∈ R be such that α, β > d , then • Bony’s Linearization Theorem: for all C ∞ function F, if a ∈ H α ( R d ) then F ( a ) − F (0) − T F ′ ( a ) a ∈ H α − d ( R d ) . • If a ∈ H α ( R d ) and b ∈ H β ( R d ) , then ab − T a b − T b a ∈ H α + β − d ( R d ) . More-over there exists a positive constant K independent of a and b such that: k ab − T a b − T b a k H α + β − d ≤ K k a k H α k b k H β . .4. Link between pseudodifferential and paradifferential symbols. As pre-sented in [14] the link between paradifferential operators and pseudodifferential oper-ators will be as follows, for a paradifferential operator a ∈ Γ m ′ ρ ( R d ) we can associatea symbol σ a ∈ S m , ( R d × R d ). All of the results presented above were for the class S m , ⊂ S m , and don’t all generalize easily. To remedy this, the idea in paradifferen-tial calculus is regularization by cut-off in the frequency domain thus σ a will havean extra spectral localization property that will give them the desired properties asin S m , . Definition-Proposition 3.1. Take m ∈ R , Σ m W ( R d ) denotes the subclass of symbols σ ∈ Γ m W ( R d ) which satisfy the following spectral condition ∃ ǫ < , F x σ ( ξ, η ) = 0 for | ξ | > ǫ ( | η | + 1) . (3.5) When W = W r, ∞ (Ω) we write Σ m W ( R d ) = Σ mr ( R d ) .When W ⊂ L ∞ ( R d ) , Γ m W ( R d ) ⊂ Γ m ( R d ) and Σ m W ( R d ) ⊂ Σ m ( R d ) . Moreover, bythe Bernstein inequalities (2.1) , Σ m ( R d ) ⊂ S m , ( R d ) . More generally, the spectralcondition implies that symbols in Σ m W ( R d ) are smooth in x too. Remark 3.4. The interesting fact now is Σ m ( R d ) still enjoys all of the symboliccalculus properties announced above for S m , ( R d ) . Definition-Proposition 3.2. (Regularization of a symbol) Take m ∈ R , a ∈ Γ m W and ψ an admissible cut-off function. Define σ ψa : F x σ a ( ξ, η ) = ψ ( ξ, η ) F x a ( ξ, η ) , thus Op( σ ψa ) = T a . Moreover for r ≥ , when W = W r, ∞ (Ω) we have the followingproperties:(1) This association is bounded: M mr ( σ ψa ) ≤ CM mr ( a ) . (2) a − σ ψa ∈ Γ m and: M m − r ( σ ψa − a ) ≤ CM mr ( a ) . In particular, if ψ and ψ are two admissible cut-off functions then thedifference σ ψ a − σ ψ a belongs to Σ m − r and: M m − r ( σ ψ a − σ ψ a ) ≤ CM mr ( a ) . Now we list a couple of important calculus properties to the association a σ ψa .For the following Proposition we consider ψ fixed and drop it from the notations. Proposition 3.9. • For m ∈ R , r ≥ , α ∈ N d of length | α | ≤ r and a ∈ Γ mr : ∂ αx σ a = σ ∂ αx a ∈ Σ m . • For m ∈ R , r ≥ and α ∈ N d of length | α | ≥ r the mapping a ∂ αx σ a isbounded from Γ mr to Σ m + | α |− r , more precisely: M m + | α |− r ( ∂ αx σ a ) ≤ M mr ( a ) . • For m ∈ R , r ≥ , β ∈ N d and a ∈ Γ mr ∂ βξ σ a − σ ∂ βξ a ∈ Σ m −| β |− r . From [14] we give an approximation of symbols in Σ m ( R d ) by symbols in theSchwartz class. Lemma 3.2. For all σ ∈ Σ m , there is a sequence of symbols σ n ∈ S ( R d × R d ) suchthat 1) the family { σ n } is bounded in S m , ,(2) the σ n satisfy the spectral condition (3.5) for some ǫ < independent of k,(3) σ n → σ on compact subsets of R d × R d . In order to give the link between Paradifferential calculus and Fourier IntegralOperators we start by defining the space of amplitudes for Paradifferential operators. Definition-Proposition 3.3. Take m ∈ R , A m W ( R d ) denotes the subclass of symbols c ∈ Γ m ( W × W × R d ) which satisfy the following spectral condition ∃ ǫ < , F x,y c ( ξ, ζ, η ) = 0 for | ξ − ζ | > ǫ ( | η | + 1) or | ζ | > ǫ ( | η | + 1) . (3.6) When W = W r, ∞ (Ω) we write A m W ( R d ) = A mr ( R d ) .By the Bernstein inequalities (2.1) , A m ( R d ) ⊂ S m , ( R d ) . More generally, the spectralcondition implies that symbols in A m W ( R d ) are smooth in x, y too. Proposition 3.10. Consider two real numbers m ∈ R , r ∈ R + and an amplitude c ∈ A mr ( R d ) , then: σ ( x, ξ ) = Z Ω × R d c ( x, y, η ) e i ( x − y ) . ( η − ξ ) dydη ∈ Σ mr ( R d ) and we have: ∀ u ∈ C ∞ (Ω) , op ( x − y ) .ξ ( c ) u = Op( σ ) u = (2 π ) − d Z R d e ix.ξ σ ( x, ξ )ˆ u ( ξ ) dξ. Moreover the asymptotic expansion of a is given by: σ ( x, ξ ) − X | α | First by Lemma 3.2 we can work with an amplitude c in S . As S ⊂ S m , by Proposition 3.7 we have σ ( x, ξ ) = Z R d × R d c ( x, y, η ) e i ( x − y ) . ( η − ξ ) dydη ∈ S . Moreover writing F x σ ( ξ, η ) = Z R d c ( ξ + η − ˜ η, ˜ η − η, ˜ η ) d ˜ η, we see that if c verifies the spectral condition with parameter ǫ then so does σ withparameter ǫ thus σ ∈ Σ mr ( R d ). The asymptotic expansion comes from the one givenin Proposition 3.7 combined by the symbolic calculus rules in Proposition 3.9. (cid:3) Pull-back of pseudo and para- differential operators Let Ω , Ω ′ be two open subsets of R d . Henceforth we will note all variables in Ω ′ with a ′ for clarity in the computations. Let χ : Ω → Ω ′ be a C ∞ map, χ gives risenaturally to the pull back operation for functions and kernels: C ∞ (Ω ′ ) → C ∞ (Ω) C ∞ (Ω ′ × Ω ′ ) → C ∞ (Ω × Ω) v v ◦ χ = v ∗ K ( x ′ , y ′ ) K ( χ ( x ) , χ ( y )) | det Dχ ( y ) | = K ∗ ( x, y ) . This Pull back has the property: K ∗ v ∗ = Z Ω K ( χ ( x ) , χ ( y )) v ( χ ( y )) | det Dχ ( y ) | dy = Z Ω ′ K ( χ ( x ) , y ′ ) v ( y ′ ) χ − ( y ′ ) dy ′ = ( K ( v χ − )) ∗ . (4.1)Where χ − : Ω ′ → ¯ N is the function counting the number if pre-images and v ∈ C ∞ (Ω ′ ). We note that the the change of variables is well defined if and only f one of the two integrals is defined. If χ is a diffeomorphism we have the usualfonctorial property K ∗ v ∗ = ( Kv ) ∗ which permits the definition of operators withkernels on manifolds.The classic result on the change of variables in pseudo-differential operators is thatfor A ∈ S mloc (Ω ′ × R d ) properly supported with kernel K then the operator definedby K ∗ is a pseudo-differential operator A ∗ of order m on Ω which is also properlysupported. Thus it can be seen as the stability of this sub-class of operators ofkernels under the pull back by diffeomorphisms (modulo a C ∞ kernel as in Remark3.2) and thus are well defined on manifolds by the same process. Before we start bypresenting those classic results we will discuss why they are essentially optimal.We start by computing for a pseudo-differential operator defined by a ∈ S m (Ω ′ × R d ) with kernel K and χ : Ω → Ω ′ a C ∞ map: K ∗ u = Z Ω K ( χ ( x ) , χ ( y )) u ( y ) | det Dχ ( y ) | dy = Z Ω × Ω (2 π ) − d e i ( χ ( x ) − χ ( y )) .ξ a ( χ ( x ) , ξ ) u ( y ) | det Dχ ( y ) | dydξ thus K ∗ = op ( χ ( x ) − χ ( y )) .ξ ( a ( χ ( x ) , ξ ) | det Dχ ( y ) | )with a ( χ ( x ) , ξ ) | det Dχ ( y ) | ∈ S m (Ω × Ω × R d ) , because and all the derivatives of χ are bounded. Put ω χ ( x, y, ξ ) = ( χ ( x ) − χ ( y )) .ξ, by the definitions on Fourier integral operators we have: C ω χ = (cid:8) ( x, y ) ∈ Ω , χ ( x ) = χ ( y ) (cid:9) . We also see that w χ is non degenerate on Ω × Ω if and only if χ is a local diffeomor-phism. To sum up: Proposition 4.1. Take a ∈ S m (Ω ′ × R d ) and χ ∈ C ∞ (Ω , Ω ′ ) . Then the pull-backof Op( a ) under χ is a Fourier Integral Operator with phase function w χ and symbol a ( χ ( x ) , ξ ) | det Dχ ( y ) | ∈ S m (Ω × Ω × R d ) . We have: C ω χ = (cid:8) ( x, y ) ∈ Ω , χ ( x ) = χ ( y ) (cid:9) . Moreover, w χ is non-degenerate if and only of χ is a local diffeomorphism. Now we ask the question if there exists a symbol a ∗ such that: op ω χ ( a ( χ ( x ) , ξ ) | det Dχ ( y ) | ) = Op( a ∗ ) . The classic result is that this true if χ is a diffeomorphism. Now we precise that it’sessentially optimal as it could be seen by the following two examples: • The necessity of the injectivity of ξ : we take χ = | | which is a local diffeo-morphism from R \ R + ∗ . We compute for A = Id i.e a = 1: op ω χ ( a ( χ ( x ) , ξ ) | det Dχ ( y ) | ) u = u ( x ) + u ( − x ) , and the part u ( . ) u ( − . ) is not a pseudo-differential operator. • The necessity of the local diffeomorphism hypothesis: we take χ = x whichis a local diffeomorphism from R \ R . We compute for A = ddx i.e a = iξ : op ω χ ( a ( χ ( x ) , ξ ) | det Dχ ( y ) | ) u = u ′ ( x )3 x , hich is a pseudo-differential operator on R \ R with a regular symbol in 0. Now we present the classic results of change of variables in pseudo and para-differential operators under the hypothesis that χ is a diffeomorphism as they canbe found in [3],[4] and [11]. Theorem 4.1. Let χ : Ω → Ω ′ be a C ∞ diffeomorphism and A = a ( x, D ) ∈ S mloc (Ω ′ × R d ) a properly supported pseudo-differential operator with kernel K.Then the operator A ∗ defined by K ∗ i.e: ∀ u ∈ C ∞ (Ω) , A ∗ u = Z Ω K ( χ ( x ) , χ ( y )) u ( y ) | det Dχ ( y ) | dy is a properly supported pseudo-differential operator with symbol a ∗ ( x, ξ ) = ( − d e − ix.ξ Z Ω × R d a ( χ ( x ) , η ) e i ( χ ( x ) − χ ( y )) .η + iy.ξ | det Dχ ( y ) | dydη ∈ S mloc (Ω × R d ) . An expansion of a ∗ is given by: a ∗ ( x, ξ ) ∼ X α α ∂ α a ( χ ( x ) , Dχ − ( χ ( x )) t ξ ) P α ( χ ( x ) , ξ ) , (4.2) where, P α ( x ′ , ξ ) = D αy ′ ( e i ( χ − ( y ′ ) − χ − ( x ′ ) − Dχ − ( x ′ )( y ′ − x ′ )) .ξ ) | y ′ = x ′ and P α is polynomial in ξ of degree ≤ | α | , with P = 1 , P = 0 . Remark 4.1. This a classic result found commonly in the literature, And as in theRemark 3.4 an analogous result still holds in the class Σ m as will be shown in theproof of the next Theorem. For para-differential operators we have: Theorem 4.2. Let χ : Ω → Ω ′ be a W ρ, ∞ loc diffeomorphism with Dχ ∈ W ρ, ∞ and ρ ≥ . Consider a ∈ Γ mr ( R d ) a properly supported paradifferential operator.Then there exists a property supported a ∗ ∈ Γ mmin ( r,ρ ) ( R d ) defined by: ( T a u ) ◦ χ = T a ∗ ( u ◦ χ ) + ( Rχ ) u, where R ∈ Γ mr ( R d ) and Rχ is a term depending essentially on χ and it’s explicitformula is given in (5.4) .Moreover a ∗ has the local expansion: a ∗ ( x, ξ ) ∼ X α | α |≤⌊ min ( r,ρ ) ⌋ α ∂ α a ( χ ( x ) , Dχ − ( χ ( x )) t ξ ) P α ( χ ( x ) , ξ ) , (4.3) where, P α ( x ′ , ξ ) = D αy ′ ( e i ( χ − ( y ′ ) − χ − ( x ′ ) − Dχ − ( x ′ )( y ′ − x ′ )) .ξ ) | y ′ = x ′ and P α is polynomial in ξ of degree ≤ | α | , with P = 1 , P = 0 . An analogous result still holds for para-differential operators modeled on thespaces a ∈ C r ∗ , r > χ ∈ C ρ ∗ .As we couldn’t find a clear reference to this result in the literature, it is eluded toin [3] , we give a simple proof of this Theorem. In fact it can be treated in the more general frame of operators with singular symbols but thisgoes beyond the scope of this work. part 3.3 point h, which can be found in pages 114-115. roof. Taking ψ a cut-off function with parameters ǫ , ǫ , and take u ∈ C ∞ (Ω)compute( T a ( u ◦ χ − )) ◦ χ = op ( χ ( x ) − χ ( y )) .ξ ( σ ψa ( χ ( x ) , ξ ) | det Dχ ( y ) | ) u = Z Ω × R d e i ( χ ( x ) − χ ( y )) · ξ σ ψa ( χ ( x ) , ξ ) | det Dχ ( y ) | u ( y ) dydξ. As we remarked above the main contribution in this integral will come from ( x, y, ξ ) ∈ C ω χ where we recall ω χ ( x, y, ξ ) = ( χ ( x ) − χ ( y )) · ξ . To show this insert the smoothcut-off function θ ( x, y ) supported in a small neighborhood of the diagonal ( x, x ).( T a ( u ◦ χ − )) ◦ χ = Z Ω × R d e i ( χ ( x ) − χ ( y )) · ξ σ ψa ( χ ( x ) , ξ ) | det Dχ ( y ) | u ( y ) dydξ = Z Ω × R d e i ( χ ( x ) − χ ( y )) · ξ θ ( x, y ) σ ψa ( χ ( x ) , ξ ) | det Dχ ( y ) | u ( y ) dydξ + Z Ω × R d e i ( χ ( x ) − χ ( y )) · ξ (1 − θ ( x, y )) σ ψa ( χ ( x ) , ξ ) | det Dχ ( y ) | u ( y ) dydξ Now ω χ has no critical points on the support of (1 − θ ( x, y )) and by integration byparts we have:( T a ( u ◦ χ − )) ◦ χ = Z Ω × R d e i ( χ ( x ) − χ ( y )) · ξ θ ( x, y ) σ ψa ( χ ( x ) , ξ ) | det Dχ ( y ) | u ( y ) dydξ + Ru. with R ∈ Γ m − min( r,ρ )0 . We now analyze when y is close to x . By the mean valueTheorem, for y sufficiently close to x , there exists a invertible linear mapping L x,y ∈ W ρ, ∞ such that ( χ ( x ) − χ ( y ) = L x,y · ( x − y ) L x,x = Dχ ( x ) . Thus we get,( T a ( u ◦ χ − )) ◦ χ = Z Ω × R d e i ( χ ( x ) − χ ( y )) · ξ θ ( x, y ) σ ψa ( χ ( x ) , ξ ) | det Dχ ( y ) | u ( y ) dydξ + Ru = Z Ω × R d e i ( x − y ) · ξ θ ( x, y ) σ ψa ( χ ( x ) , L tx,y − ξ ) | det Dχ ( y ) | (cid:12)(cid:12) det L − x,y (cid:12)(cid:12) u ( y ) dydξ + Ru. We get an operator with an amplitude c ( x, y, ξ ) = θ ( x, y ) σ ψa ( χ ( x ) , L tx,y − ξ ) | det Dχ ( y ) | (cid:12)(cid:12) det L − x,y (cid:12)(cid:12) ∈ Γ mρ ( R d ) . In the frequency domain this amplitude depends on terms coming from σ ψa ( χ ( x ) , L tx,y − ξ ), | det Dχ ( y ) | and (cid:12)(cid:12) det L − x,y (cid:12)(cid:12) . Putting all of the high frequency terms depending on χ and χ − in a term Rχ by defining ˜ ψ as the cut-off function in both of the variables( x − y, y ) with parameters: c = min(1 , sup Dχ − , sup Dχ ) , ˜ ǫ = ǫ c , ˜ ǫ = ǫ c . Thus by Proposition 3.2˜ c ( x, y, ξ ) = ˜ ψ ( D, · ) c ∈ Σ mmin ( r,ρ ) ( R d ) , with c = ˜ c + Rχ + R ′ nd R ′ ∈ Γ m − min ( r,ρ )0 . The result then follows from Proposition 3.10. (cid:3) Paracomposition Main results for paracomposition on R d . We start by a formal computa-tion, as in [18], using the Littlewood-Paley decomposition and two functions u and χ : u ◦ χ = X k ≥ u (Φ k +1 χ ) − u (Φ k χ ) = X j,k u j (Φ k +1 χ ) − u j (Φ k χ )= X j Heuristically the term 1 has frequencies of u smaller than that of χ and as in classical paradifferential results will depend mainly on the regularity of χ .This is indeed the main term in Bony’s para-linearization Theorem modulo a moreregular remainder: (1) = X k ≥ (cid:18) Z Φ k − u ′ ( τ Φ k χ + (1 − τ )Φ k − ) χdτ (cid:19) φ k χ = X k ≥ Φ k − ( u ′ ◦ χ )( φ k χ ) | {z } T u ′◦ χ χ (5.2)+ X k ≥ (cid:18) Z Φ k − u ′ ( τ Φ k χ + (1 − τ )Φ k − χ ) − Φ k − ( u ′ ◦ χ ) dτ (cid:19) φ k χ | {z } R . Same as term 1, heuristically term 2 will essentially depend on the regularity of u,with a remainder depending on χ and u that is more regular when it’s well defined.Thus (2) will naturally give rise to the paracomposition operator. To better under-stand it, let us suppose just for the next computation that χ is linear and invertible: (2) = X k ≥ Z R d φ k ( ξ )ˆ u ( ξ ) e i Φ k χ ( x ) .ξ dξ = X k ≥ Z R d φ k (Φ k χ − t ξ )ˆ u (Φ k χ − t ξ ) e ix.ξ | Φ k χ − t ( ξ ) | dξ Thus we essentially have to look at how Φ k χ − t modifies the frequencies and thus howit modifies the rings in the Littlewood-Paley decomposition.Put (cid:8) k ≥ , C ′ k = supp φ k (Φ k χ − t . ) (cid:9) , we have: C ′ k ≈ [ k − N ′ ≤ l ≤ k + N C l , where N and N’ are such that N > sup k, R d | Φ k χ ′ | and N ′ > sup k, R d | Φ k χ ′ | − and the natural para-composition operator in this case is obtained by cutting thefrequencies according to C ′ k , this is exactly the ”lemme de recoupe” in Alinhac’swork. ow we define N as in the previous remark and compute:(2) = X k ≥ X l ≥ l ≤ k + N φ l ( D )( u k ◦ χ | {z } χ ⋆ u ) (5.3)+ X k ≥ X l ≥ l ≤ k + N φ l ( D )[ u k ◦ Φ k χ − u k ◦ χ ] | {z } R + X k ( Id − Φ k + N )( D ) u k ◦ Φ k χ | {z } R . Theorem 5.1. Let χ : R d → R d be a C ρ ∗ ,loc map with Dχ ∈ C ρ ∗ and ρ > . Thenfor all σ, s ∈ R ∗ + the following maps are continuous: C σ ∗ ( R d ) → C σ ∗ ( R d ) C σ ∗ ,loc ( R d ) → C σ ∗ ,loc ( R d ) u χ ⋆ u = X k ≥ X l ≥ l ≤ k + N φ l ( D ) u k ◦ χ u χ ⋆ u = X k ≥ X l ≥ l ≤ k + N φ l ( D ) u k ◦ χ. If moreover χ is a diffeomorphism then we have the Sobolev estimates: H s ( R d ) → H s ( R d ) H sloc ( R d ) → H sloc ( R d ) u χ ⋆ u = X k ≥ X l ≥ l ≤ k + N φ l ( D ) u k ◦ χ u χ ⋆ u = X k ≥ X l ≥ l ≤ k + N φ l ( D ) u k ◦ χ. Taking ˜ χ : R d → R d a C ρ ∗ ,loc map with D ˜ χ ∈ C ˜ ρ ∗ and ˜ ρ > , then the previousoperation has the natural fonctorial property: ∀ u ∈ C σ ∗ ( R d ) ∪ C σ ∗ ,loc ( R d ) , χ ⋆ ˜ χ ⋆ u = ( χ ◦ ˜ χ ) ⋆ u + Ru. with R, R : C σ ∗ ( R d ) → C σ + min ( ρ, ˜ ρ ) ∗ ( R d ) , R : C σ ∗ ,loc ( R d ) → C σ + min ( ρ, ˜ ρ ) ∗ ,loc ( R d ) , and if χ and ˜ χ are diffeomorphisms: R : H s ( R d ) → H s + min ( ρ, ˜ ρ ) ( R d ) , R : H sloc ( R d ) → H s + min ( ρ, ˜ ρ ) loc ( R d ) . Remark 5.2. It’s natural that the Sobolev estimates only hold when χ is a diffeo-morphism because for example even the usual composition operation u u ◦ χ is notnecessarily continuous on L p spaces, p < ∞ . An extra hypothesis that appears inthe literature is χ is a local diffeomorphism with all of it’s local inverses uniformlybounded in ˙ W , ∞ . Theorem 5.2. Let u be a W , ∞ ( R d ) map and χ be a C ρ ∗ ,loc map with Dχ ∈ C ρ ∗ and ρ > . Then: u ◦ χ ( x ) = χ ⋆ u ( x ) + T u ′ ◦ χ χ ( x ) + R ( x ) + R ( x ) + R ( x ) where the paracomposition given in the previous Theorem verifies the estimates: ∀ σ > , k χ ⋆ u ( x ) k σ ≤ C ( k Dχ k ∞ ) k u ( x ) k σ ,u ′ ◦ χ ∈ Γ W , ∞ ( R d ) ( R d ) f or u Lipchitz,and the remainders verify the estimates: • In Zygmund Spaces, for σ > : k R k ρ + min (1+ ρ,σ ) ≤ C k Dχ k ρ k u k σ f or i ∈ { , } , k R i k ρ + σ ≤ C ( k Dχ k ∞ ) k Dχ k ρ k u k σ . Clearly when there is no diffeomorphism hypothesis on χ we can choose χ : R d → R d ′ with d = d ′ and have the same results but for clarity we chose to present the same dimensions in thispresentation. In Sobolev Spaces, for s > d we get the following estimates – without the diffeomorphism hypothesis: k R k H ρ + min (1+ ρ,s − d ≤ C k Dχ k ρ k u k H s k R k H ρ + s ≤ C ( k Dχ k ∞ ) k Dχ k ρ k u k H s . – Suppose moreover that χ is a diffeomorphism: k R k H ρ + s ≤ C ( k Dχ k ∞ , (cid:13)(cid:13) Dχ − (cid:13)(cid:13) ∞ ) k Dχ k ρ k u k H s . The same estimates hold in the local spaces. As Alinhac remarked in [3], a particular case of the previous Theorem is Bonypara-linearization Theorem but with the extra hypothesis of diffeomorphism, hereit’s is a full generalization because we dropped the diffeomorphism hypothesis. Wefind Bony’s para-linearization Theorem when σ = + ∞ , in this case only the term T u ′ ◦ χ χ ( x ) appears and χ ⋆ u ( x ) is a part of the remainder. If on the other hand, χ ∈ C ∞ , the term T u ′ ◦ χ χ ( x ) becomes a part of the remainder and the paracomposition χ ⋆ u ( x ) coincides with the usual composition modulo a regularizing operator. ThusTheorem 5.2 appears as a linearization Theorem of u ◦ χ as the sum of two terms, onedepending mainly on the regularity of u (and ”less” of χ ) and the other dependingmainly on the regularity of χ (and ”less” of u ).The simplest example here is when χ ( x ) = Ax is a linear operator and in thatcase we see that: u ( Ax ) ∼ ( Ax ) ∗ u, and T u ′ ( Ax ) Ax ∼ . Remark 5.3. The proof of Theorem 5.2 tell us that the if in the sum defining χ ⋆ wechoose a different N ′ ≥ N then the operator is modified by a ρ regularizing operator. Theorem 5.3. Consider a ∈ Γ mβ ( R d ) , with β ≥ , χ : R d → R d a C ρ ∗ ,loc map with Dχ ∈ C ρ ∗ , ρ > and ρ / ∈ N . Then there exists q ∈ Γ m − β ( R d ) such that we havethe following formal symbolic calculus rule: χ ⋆ T a u = op ω χ (cid:18) σ a ( χ ( x ) , ξ ) | det Dχ ( y ) | χ − ( χ ( y )) (cid:19) χ ⋆ u + op ω χ (cid:18) σ q ( χ ( x ) , ξ ) | det Dχ ( y ) | χ − ( χ ( y )) (cid:19) χ ⋆ u. To join Alinhac’s work, the following Proposition makes the link between hisdefinition of the paracomposition operator in the case of a diffeomorphism and theone given here. Theorem 5.4. Let u be W , ∞ ( R d ) map and χ be a C ρ ∗ ,loc diffeomorphism with Dχ ∈ C ρ ∗ and ρ > . Consider ˜ N such that ˜ N > sup k, R d | Φ k χ ′ | − and ˜ N > sup k, R d | Φ k χ ′ | .Put Alinhac’s paracomposition operator: χ ∗ u = X k ≥ X l ≥ | l − k |≤ ˜ N φ l ( D ) u k ◦ χ then: χ ∗ u = χ ⋆ u + R , Where the remainder verifies: • In Zygmund Spaces, for σ > : k R k ρ + σ ≤ C ( (cid:13)(cid:13) Dχ − (cid:13)(cid:13) ∞ , k Dχ k ∞ ) k Dχ k ρ k u k σ . • In Sobolev Spaces, for s > d : k R k H ρ + s ≤ C ( (cid:13)(cid:13) Dχ − (cid:13)(cid:13) ∞ , k Dχ k ∞ ) k Dχ k ρ k u k H s . he same estimates hold in the local spaces.Take a ∈ Γ mβ ( R d ) and q as in Theorem 5.3 then: χ ⋆ T a u = T a ∗ χ ⋆ u + T q ∗ χ ⋆ uχ ∗ T a u = T a ∗ χ ∗ u + T q ′∗ χ ∗ u with q ′ ∈ Γ m − β ( R d ) . Remark 5.4. As in remark 5.3, the proof of Theorem 5.4 tell us that the if in thesum defining χ ∗ we choose a different ˜ N ′ ≥ ˜ N then the operator is modified by a ρ regularizing operator. Remark 5.5. As a corollary of Theorem 5.4 we get that in Theorem 4.2: Rχ = T ( T a u ) ′ ◦ χ χ − T a ∗ T u ′ ◦ χ χ. (5.4)5.2. Proofs. We will give the proof for the estimates in global spaces, for localspaces it is sufficient to see that the given estimates hold under the hypothesis thatall the functions used have a compact support and to pass to local spaces estimatesit is sufficient to multiply by functions in C ∞ which don’t modify the estimates given(we don’t make any boundary estimates). Proof of Theorem 5.1 and 5.2. Take χ : R d → R d be a C ρ ∗ map with ρ > , N + 1).We start by the Zygmund spaces estimates (thus we don’t suppose that χ is adiffeomorphism): k Φ k + N u k ◦ χ k ∞ ≤ C k u k k ∞ ≤ − kσ k u k σ and supp Φ k + N u k ◦ χ ⊂ k B.Thus by Proposition 2.3, for σ > χ ⋆ u ∈ C σ ∗ ( R d ) and k χ ⋆ u k σ ≤ C ( N )1 − − σ k u k σ . For Sobolev estimates we suppose that χ is a diffeomorphism and by the change ofvariables formula we have for s > k Φ k + N u k ◦ χ k L ≤ C ( (cid:13)(cid:13) Dχ − (cid:13)(cid:13) ∞ ) k u k k L ≤ C ( (cid:13)(cid:13) Dχ − (cid:13)(cid:13) ∞ )2 − ks k u k H s and supp Φ k + N u k ◦ χ ⊂ k B.Thus by Proposition 2.3, for σ > χ ⋆ u ∈ H s ( R d ) and k χ ⋆ u k H s ≤ C ( N, (cid:13)(cid:13) Dχ − (cid:13)(cid:13) ∞ )1 − − s k u k H s . Now we compute the estimates on the remainders in the linearization formula. R = X k ≥ (cid:18) Z Φ k − u ′ ( τ Φ k χ + (1 − τ )Φ k − χ ) − Φ k − ( u ′ ◦ χ ) dτ (cid:19) φ k χ = X k r k χ k r k = Z Φ k − u ′ ( τ Φ k χ + (1 − τ )Φ k − χ ) − Φ k − ( u ′ ◦ χ ) dτ = Z Z Φ k − u ′′ ( tτ (Φ k χ − Φ k − χ )+ t (Φ k − χ − χ ) − χ ) dt [ τ (Φ k χ − χ ) + (1 − τ )(Φ k − χ − χ )] dτ. Thus if σ = 1: (cid:13)(cid:13) r k (cid:13)(cid:13) ∞ ≤ C k ( − σ − ρ ) And if σ = 1: (cid:13)(cid:13) r k (cid:13)(cid:13) ∞ ≤ Ck k ( − − ρ ) ≤ C − k , hich sums up in (cid:13)(cid:13) r k (cid:13)(cid:13) ∞ ≤ C − min (1+ ρ,σ ) k . By the same computations we haveanalogous estimates on (cid:13)(cid:13) ∂ α r k (cid:13)(cid:13) and clearly r k ∈ C ∞ which gives the desired esti-mates on R by Lemma 2.1 and the fact that r = 0, both in the Sobolev et Zygmundcases without the diffeomorphism hypothesis. R = X k ≥ φ k + N ( D )[ u k ◦ Φ k χ − u k ◦ χ ]= X k ≥ φ k + N ( D )[( Z u ′ k ( t Φ k + (1 − t ) χ ) dt )(Φ k χ − χ )]= X k ≥ φ k + N ( D )[ r k (Φ k χ − χ )] . We have: (cid:13)(cid:13) r k (cid:13)(cid:13) ∞ ≤ C − kσ combining this with Propositions 2.3, 2.4 and the fact that r = 0 we get the desiredestimates again in both in the Sobolev et Zygmund cases without the diffeomor-phism hypothesis.The proof of the estimates on R relies on oscillatory integral techniques that comefrom Lemma 3.1. For the sake of completion we will give the explicit computationswithout directly using the Lemma. R ( x ) = X k ( Id − Φ k + N )( D ) u k ◦ Φ k χ ( x ) . We will prove that for j ≥ k + N + 1 , ν ≥ ρ > 0, we have: k φ j ( D ) u k ◦ Φ k χ k ∞ ≤ C ν ( k Dχ k ρ )2 − jν k ( ν − ρ ) k u k k ∞ (5.5)which will be sufficient to give the Zygmund estimates on R because we will have: k φ j ( D ) R k ∞ ≤ X k ≥ k ≤ N − j +1 k φ j ( D ) u k ◦ Φ k χ k ∞ ≤ X k ≥ k ≤ N − j +1 C ν ( k Dχ k ρ )2 − jν k ( ν − ρ ) k u k k ∞ ≤ X k ≥ k ≤ N − j +1 C ν ( k Dχ k ρ )2 − jν k ( ν − ρ ) k u k k ∞ ≤ X k ≥ k ≤ N − j +1 C ν ( k Dχ k ρ )2 − jν k ( ν − ρ − σ − k u k σ , Taking ν > ρ + σ we dominate the last expression by: C ν ( k Dχ k ρ )2 − j ( ρ + σ +1) k u k σ which gives the desired Zygmund estimate.For the Sobolev estimates we will prove that: k φ j ( D ) u k ◦ Φ k χ k ≤ C ν ( k Dχ k ρ )2 − jν k ( ν − ρ ) k u k ◦ Φ k χ k , (5.6)which then necessitates the diffeomorphism hypothesis on χ to have: k φ j ( D ) u k ◦ Φ k χ k ≤ C ν ( k Dχ k ρ , (cid:13)(cid:13) Dχ − (cid:13)(cid:13) ∞ )2 − jν k ( ν − ρ ) k u k k , nd the desired estimates follow exactly as in the Zygmund case.Now we prove (5.5) and (5.6),to make the desired estimates we will put in a testfunction f ∈ C ∞ b as it’s usually done with oscillatory integral estimates: φ j ( D ) f u k ◦ Φ k χ ( x ) = Z e i ( x − y ) .ξ φ j ( ξ ) φ k ( η ) f ( y ) ˆ u k ( η ) e i Φ k χ ( y ) .η dηdydξ (5.7)Set ω k ( y, η, ξ ) = Φ k χ ( y ) .η − y.ξ,L k ( y, η, ξ, ∂ y ) = Φ k χ ′ ( y ) t .η − y.ξi | Φ k χ ′ ( y ) t .η − y.ξ | . ∇ y . Given the definition of N we have: (cid:12)(cid:12) Φ k χ ′ ( y ) t .η − y.ξ (cid:12)(cid:12) ≥ C ( | η | + | ξ | ) on supp φ j ( ξ ) φ k ( η ) , Thus L k is well defined and regular, moreover L k e iω k = e iω k . Integrating by partsin (5.6): φ j ( D ) f u k ◦ Φ k χ ( x ) = Z e ix.ξ φ j ( ξ ) φ k ( η ) ˆ u k ( η ) e iw k ( L tk ) ν f ( y ) dηdydξ. Note that ( L tk ) ν f is homogeneous with degree − ν in ( η, ξ ), and smooth on thesupport of φ j ( ξ ) φ k ( η ). Also (cid:12)(cid:12) ( L tk ) ν f ( y ) (cid:12)(cid:12) ≤ C ( k f k ν , k Dχ k ρ )2 ν − σ on | ξ | + | η | = 1 . (5.8)Next on a box containing supp φ j ( ξ ) φ k ( η ), write( L tk ) ν f ( y ) = X ( α,β ) ∈ Λ a kναβ ( y ) e iα.ξ + iβ.η = 2 − jν X ( α,β ) ∈ Λ a kναβ ( y ) e i − j α.ξ + i − j β.η , where Λ is an appropriate lattice and X ( α,β ) ∈ Λ k a kναβ k ∞ ≤ C ( k f k ν , k Dχ k ρ )2 ν − σ . (5.9)So (5.7) becomes for j ≥ φ j ( D ) f u k ◦ Φ k χ ( x ) (5.10)= 2 − jν X ( α,β ) ∈ Λ Z e ix.ξ φ j ( ξ ) φ k ( η ) ˆ u k ( η ) e iw k a kναβ ( y ) e i − j α.ξ + i − j β.η dηdydξ = 2 − jν X ( α,β ) ∈ Λ Z e i ( x − y ) .ξ φ j ( ξ ) u k (Φ k χ ( y ) + 2 − j β ) a kναβ ( y ) e i − j α.ξ dydξ = 2 − jν X ( α,β ) ∈ Λ Z e i ( x − y ) .ξ jn ˆ φ (2 j ( x − y ) + α ) u k (Φ k χ ( y ) + 2 − j β ) a kναβ ( y ) dy. = 2 − jν X ( α,β ) ∈ Λ ( a kναβ · u k (Φ k χ + 2 − j β )) ∗ g α ( x ) , (5.11)Where g α ( x ) = 2 jn ˆ φ (2 j x + α ) thus k g α k L = 2 jn Z (cid:12)(cid:12)(cid:12) ˆ φ (2 j x + α ) (cid:12)(cid:12)(cid:12) dx = (cid:13)(cid:13)(cid:13) ˆ φ (cid:13)(cid:13)(cid:13) L . (5.12)For j = 0 we have an analog inequality.Using the classic Young and H¨older inequalities combined with (5.9), (5.12) andtaking f → roof of Theorem 5.3. Take a ∈ Γ mβ ( R d ), with β ≥ χ : R d → R d a C ρ ∗ mapwith ρ > 0. We compute: χ ⋆ T a u = X k ≥ Φ k + N [( T a u ) k ◦ χ ] , (5.13)Note that ( T a u ) k can been as T φ k T a u and seeing this a modification of the cut-offfunction by Proposition 3.2 we get:( T a u ) k = T φ k T a u = T a T φ k u + T q k u, with q k ∈ Γ m − β ( R d ) . Put q = P q k then (5.13) becomes: χ ⋆ T a u = X k ≥ Φ k + N [( T a u k ) ◦ χ ] + X k ≥ Φ k + N [( T q k u k ) ◦ χ ] . And the formal discussion and computations in part 4 give the desired result. Proof of Theorem 5.4. The only thing left to prove is the estimate on R . R = X k Φ k − ˜ N ( D ) u k ◦ Φ k χ ( x ) | {z } + φ N ( D ) u k ◦ χφ N ( D ) u k ◦ χ is C ∞ so we only have to the estimate to the first term on the left handside. Estimating 1 is exactly as (5.7) but with φ j substituted by Φ k − ˜ N . The coreof the estimation relies on the fact that L k should be well defined and regular onsupp Φ k − ˜ N ( ξ ) φ k ( η ) which is the case give our choice of ˜ N and the fact k ≥ 1. Wealso have the estimate: (cid:12)(cid:12) Φ k χ ′ ( y ) t .η − y.ξ (cid:12)(cid:12) ≥ C ( | η | + | ξ | ) on supp Φ k − ˜ N ( ξ ) φ k ( η ) . The proof than exactly follows as for R .5.3. Main results for paracomposition on open subsets. The previous defini-tion of the operator χ ⋆ on functions defined on R d relied heavily on the Littelwood-Paley theory which doesn’t make it immediately extendable to the open domaincase. In [3], Alinhac was able to define such an operator profiting from the continu-ity of χ ∗ on the local function spaces and a partition of unity on the open domains.More precisely consider ( V i , Θ i ) a partition of unity locally finite of Ω ′ then: u ◦ χ = X i Θ i u ◦ χ where Θ i u is seen as a function of R n with the natural extension by 0. In order tohave the same natural extension for χ , χ − (supp Θ i )needs to be compact we thus have to suppose that χ is a proper map . Under thishypothesis consider ζ i ∈ C ∞ (Ω) such that ζ i = 1 on χ − (supp Θ i ): u ◦ χ = X i ζ i Θ i u ◦ ζ i χ, (5.14)where ζ i χ is seen as a function of R n with the natural extension by 0. Note that this extra hypothesis is needed for the methods used to work and is not intrinsic tothe problem. Also this hypothesis is immediately verified in the diffeomorphism case treated byAlinhac. heorem 5.5. Let χ : Ω → Ω ′ be a C ρ ∗ ,loc proper map with Dχ ∈ C ρ ∗ and ρ > . Consider ( V i , Θ i ) a partition of unity locally finite of Ω ′ and ζ i the associatedfunctions as previously. Then for all σ, s ∈ R ∗ + the following maps are continuous: C σ ∗ (Ω ′ ) → C σ ∗ (Ω) C σ ∗ ,loc (Ω ′ ) → C σ ∗ ,loc (Ω) u χ ⋆ u = X i ζ i · ( ζ i χ ) ⋆ Θ i u u χ ⋆ u = X i ζ i · ( ζ i χ ) ⋆ Θ i u, if moreover χ is a diffeomorphism then we have the Sobolev estimates: H s (Ω ′ ) → H s (Ω) H sloc (Ω ′ ) → H sloc (Ω) u χ ⋆ u = X i ζ i · ( ζ i χ ) ⋆ Θ i u u χ ⋆ u = X i ζ i · ( ζ i χ ) ⋆ Θ i u, where Θ i u and ζ i χ are treated as functions on R d . And In the sum defining each ( ζ i χ ) ⋆ a choice N i , N i ≥ sup supp Θ i χ ′ is made by the definition in section 5.1, but by remark 5.3 in order to simplify thecomputations we can take the same N ≥ N i , N ≥ sup Ω χ ′ uniformly for all the operators and this modifies the definition by a ρ regularizingoperator.Making a different choice ( V ′ i , Θ ′ i , ζ ′ i ) , which gives a different operator χ ⋆ then ∀ u ∈ C σ ∗ ( R d ) ∪ C σ ∗ ,loc ( R d ) , χ ⋆ u = χ ⋆ u + R ′ u. with R ′ u ∈ C ∞ .Consider ˜ χ : Ω ′ → Ω ′′ a a C ρ ∗ ,loc proper map with D ˜ χ ∈ C ˜ ρ ∗ with ˜ ρ > , then theprevious operation has the natural fonctorial property: ∀ u ∈ C σ ∗ (Ω ′′ ) ∪ C σ ∗ ,loc (Ω ′′ ) , χ ⋆ ˜ χ ⋆ u = ( χ ◦ ˜ χ ) ⋆ u + ˜ Ru. with ˜ R, ˜ R : C σ ∗ (Ω ′′ ) → C σ + min ( ρ, ˜ ρ ) ∗ (Ω) , ˜ R : C σ ∗ ,loc (Ω ′′ ) → C σ + min ( ρ, ˜ ρ ) ∗ ,loc (Ω) , and if χ and ˜ χ are diffeomorphisms: ˜ R : H s (Ω ′′ ) → H s + min ( ρ, ˜ ρ ) (Ω) , ˜ R : H sloc (Ω ′′ ) → H s + min ( ρ, ˜ ρ ) loc (Ω) . Theorem 5.6. Let u be a W , ∞ (Ω) map and χ be a be a C ρ ∗ ,loc proper map with Dχ ∈ C ρ ∗ and ρ > . Then: u ◦ χ ( x ) = χ ⋆ u ( x ) + T u ′ ◦ χ χ ( x ) + R ( x ) + R ( x ) + R ( x ) where the paracomposition given in the previous Theorem verifies the estimates: ∀ σ > , k χ ⋆ u ( x ) k σ ≤ C ( k Dχ k ∞ ) k u ( x ) k σ ,u ′ ◦ χ ∈ Γ W , ∞ (Ω) ( R d ) for u Lipchitz.The remainders are given by: R = X i X k ≥ ζ i (cid:18) Z Φ k − Θ i u ′ ( τ Φ k ζ i χ +(1 − τ )Φ k − ζ i χ ) − Φ k − (Θ i u ′ ◦ ζ i χ ) dτ (cid:19) φ k ζ i χ,R = X i X k ≥ X l ≥ l ≤ k + N ζ i ( φ l ( D )[Θ i u k ◦ Φ k ζ i χ − Θ i u k ◦ ζ i χ ]) ,R = X i X k ζ i (( Id − Φ k + N )( D )Θ i u k ◦ Φ k ζ i χ ) , nd the remainders verify the estimates: • In Zygmund Spaces, for σ > : k R k ρ + min (1+ ρ,σ ) ≤ C k Dχ k ρ k u k σ f or i ∈ { , } , k R i k ρ + σ ≤ C ( k Dχ k ∞ ) k Dχ k ρ k u k σ . • In Sobolev Spaces, for s > d we get the following estimates – without the diffeomorphism hypothesis: k R k H ρ + min (1+ ρ,s − d ≤ C k Dχ k ρ k u k H s k R k H ρ + s ≤ C ( k Dχ k ∞ ) k Dχ k ρ k u k H s . – Suppose moreover that χ is a diffeomorphism: k R k H ρ + s ≤ C ( k Dχ k ∞ , (cid:13)(cid:13) Dχ − (cid:13)(cid:13) ∞ ) k Dχ k ρ k u k H s . The same estimates hold in the local spaces. Theorem 5.7. Consider a ∈ Γ mβ ( R d ) , with β ≥ and χ : Ω → Ω ′ a C ρ ∗ ,loc propermap with Dχ ∈ C ρ ∗ , ρ > and ρ / ∈ N . Then there exists q ∈ Γ m − β ( R d ) such thatwe have the following formal symbolic calculus rule: χ ⋆ T a u = op ω χ (cid:18) σ a ( χ ( x ) , ξ ) | det Dχ ( y ) | χ − ( χ ( y )) (cid:19) χ ⋆ u + op ω χ (cid:18) σ q ( χ ( x ) , ξ ) | det Dχ ( y ) | χ − ( χ ( y )) (cid:19) χ ⋆ u. Again to join Alinhac’s work: Theorem 5.8. Let u be W , ∞ (Ω) map and χ be a W , ∞ diffeomorphism, a C ρ ∗ ,loc proper map with Dχ ∈ C ρ ∗ and ρ > . Again, consider ( V i , Θ i ) a partition ofunity locally finite of Ω ′ and ζ i the associated functions as previously. Put Alinhac’sparacomposition operator: χ ∗ u = X i ζ i ( ζ i χ ) ∗ Θ i u then : χ ∗ u = χ ⋆ u + R , Where the remainder verifies: • In Zygmund Spaces, for σ > : k R k ρ + σ ≤ C ( (cid:13)(cid:13) Dχ − (cid:13)(cid:13) ∞ , k Dχ k ∞ ) k Dχ k ρ k u k σ . • In Sobolev Spaces, for s > d : k R k H ρ + s ≤ C ( (cid:13)(cid:13) Dχ − (cid:13)(cid:13) ∞ , k Dχ k ∞ ) k Dχ k ρ k u k H s . The same estimates hold in the local spaces.Consider a ∈ Γ mβ ( R d ) and q as in Theorem 5.3 then: χ ⋆ T a u = T a ∗ χ ⋆ u + T q ∗ χ ⋆ uχ ∗ T a u = T a ∗ χ ∗ u + T q ′∗ χ ∗ u with q ′ ∈ Γ m − β ( R d ) . Again we have the same ”independence” of the definition of the operator χ ∗ (moduloa more regular term) with respect to the arbitrary choices made, more precisely,making a different choice ( V ′ i , Θ ′ i , ζ ′ i ) which gives a different operator χ ∗ then ∀ u ∈ C σ ∗ ( R d ) ∪ C σ ∗ ,loc ( R d ) , χ ∗ u = χ ∗ u + R ′ u. with R ′ u ∈ C ∞ . .4. Proof. All of the estimates given come directly for the Theorems of section5.1. The linearization formulas come from Equation (5.14) and the linearizationTheorems in section 5.1. The only thing left to prove is the independency resultwith respect to the choice of ( V i , Θ i , ζ i ). We start by the following Lemma: Lemma 5.1. Let (Θ , ζ, ˜ ζ ) ∈ C ∞ (Ω ′ ) be such that ζ = 1 on χ − (supp Θ) and ˜ ζ = 1 on supp ζ then: X k ≥ ζ Φ k + N ( D )[(Θ u ) k ◦ ζχ ] = X k ≥ ˜ ζ Φ k + N ( D )[(Θ u ) k ◦ ˜ ζχ ] + F, F ∈ C ∞ Proof. Take Θ ′ ∈ C ∞ (Ω ′ ) such that Θ ′ ◦ χ = 0 on supp ζ and Θ ′ ◦ χ = 1 on supp ˜ ζ − ζ and compute: X k ≥ ζ Φ k + N ( D )[(Θ u ) k ◦ ζχ ]= X k ≥ ˜ ζ Φ k + N ( D )[(Θ u ) k ◦ ˜ ζχ ] + X k ≥ ( ζ − ˜ ζ )Φ k + N ( D )[(Θ u ) k ◦ ˜ ζχ ]= X k ≥ ˜ ζ Φ k + N ( D )[(Θ u ) k ◦ ˜ ζχ ] + X k ≥ ( ζ − ˜ ζ )Φ k + N ( D )[(Θ ′ (Θ u ) k ) ◦ ˜ ζχ ] | {z } F . And we have by integration by parts, ∀ l ∈ N :Θ ′ (Θ u ) k = 2 − kl Z e i ( x ′ − y ′ ) ξ i ( x ′ − y ′ ) l Θ ′ ( x )Θ( y ) φ (2 − k ξ ) u ( y ) dydξ, thus , (cid:13)(cid:13) Θ ′ (Θ u ) k (cid:13)(cid:13) ∞ ≤ C l − k ( l − n ) , and F ∈ C ∞ . (cid:3) Given (i,j) such that supp Θ i ∩ supp Θ ′ j = ∅ we define ˜ ζ i,j ∈ C ∞ (Ω) such that˜ ζ i,j = 1 on supp ζ i ∪ supp ζ ′ j . χ ⋆ u = X i ζ i · ( ζ i χ ) ⋆ Θ i u = X k ≥ X i,j ζ i Φ k + N ( D )[(Θ i Θ ′ j u ) k ◦ ζ i χ ]= X k ≥ X i,j ˜ ζ i,j Φ k + N ( D )[(Θ i Θ ′ j u ) k ◦ ˜ ζ i,j χ ] + F, F ∈ C ∞ = X k ≥ X i,j ζ ′ j Φ k + N ( D )[(Θ i Θ ′ j u ) k ◦ ζ ′ j χ ] + F + F ′ , F ′ ∈ C ∞ = χ ⋆ u + F + F ′ , which gives the desired result and ends the proof. References [1] T. Alazard, N. Burq, C. Zuily: On the water waves equations with surface tension , Duke Math.J. 158(3), 413-499 (2011).[2] T. Alazard, N. Burq, C. Zuily: On the Cauchy problem for gravity water waves , Invent. Math.,198 (2014), 71-163.[3] S. Alinhac: Paracomposition et operateurs paradifferentiels , Communications in Partial Differ-ential Equations, 1986,11:1, 87-121.[4] S. Alinhac and P. G´erard: Pseudodifferential Operators and the Nash-Moser Theorem , GraduateStudies in Mathematics Volume: 82, 2007. 5] M. Bauer, M. Bruveris, E. Cismas, J. Escher, B. Kolev.: Well-posedness of the EPDiff equationwith a pseudo-differential inertia operator. Calcule symbolique et propagation des singularit´es pour les ´equations aux d´eriv´eespartielles non-lin´eaires , Ann. Scient. de l’Ecole Norl. Sup., 14(1981), 209-246.[7] J-M. Bony: Propagation des singularit´es pour les ´equations aux d´eriv´es partielles non-lin´eaires ,Sem. Goulaouic-Meyer-Schwartz, 1979-80, n22.[8] J-M. Bony: Interaction des singularit´es pour les ´equations aux d´erives partielles non-lin´eaires ,Sem. Goulaouic-Meyer-Schwartz, 1981-82, n2.[9] J-M. Bony: Interaction des singularit´es pour les ´equations de Klein-Gordon non-lin´eaires , Sem.Goulaouic-Meyer-Schwartz, 1983-84, n10.[10] S. N. Chandler-Wilde, D. P. Hewett, A. Moiola: Sobolev spaces on non-Lipschitz subsets of R n with application to boundary integral equations on fractal screens , arXiv:1607.01994, 2017.[11] L. H¨ormander: Fourier integral operators. I , Acta Math. 127 (1971), 79-183.[12] H. Inci, T. Kappeler, and P. Topalov. On the Regularity of the Composition of Diffeomor-phisms , volume 226 of Memoirs of the American Mathematical Society.[13] E. Leichtnam: Front d’onde d’une sous vari´et´e; propagation des singularit´es pour des ´equationsaux d´eriv´ees partielles non lin´eaires , Th`ese de 3´eme cycle, Universit´e Paris XI, Orsay.[14] G. M´etivier: paradifferential calculus and applications to the Cauchy problem for non linearsystems , Ennio de Giorgi Math. res. Center Publ., Edizione della Normale, 2008.[15] Y. Meyer: Remarques sur un th´eo`eme de J.M. Bony , Suppl. Rend. Circ. Mat. Palermo , n ◦ A geometric proof to the Quasi-linearity of the Water-Waves system and theincompressible Euler equations , Preprint.[17] A. Shnirelman: Microglobal Analysis of the Euler Equations , J. math. fluid mech. (2005)7(Suppl 3): S387. https://doi.org/10.1007/s00021-005-0167-5 .[18] M. E. Taylor: Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, andLayer Potentials , American Mathematical Soc., 2007., American Mathematical Soc., 2007.