Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale
GGlobal smoothness of quasiconformal mappings in theTriebel-Lizorkin scale
Kari Astala ∗ Mart´ı Prats † Eero Saksman ‡ January 24, 2019
Abstract
We give sufficient conditions for quasiconformal mappings between simply connected Lips-chitz domains to have H¨older, Sobolev and Triebel-Lizorkin regularity in terms of the regularityof the boundary of the domains and the regularity of the Beltrami coefficients of the mappings.The results can be understood as a counterpart for the Kellogg-Warchawski Theorem in thecontext of quasiconformal mappings.
Given domains Ω , Ω Ă C and µ P L p C q with } µ } L ă p µ q Ă Ω , we say that g : Ω Ñ Ω is a µ -quasiconformal mapping from Ω to Ω if it is a homeomorphism between bothdomains, it is orientation-preserving, g P W , loc p Ω q and g satisfies the Beltrami equation¯ B g “ µ B g almost everywhere. (1.1)We say that µ is the Beltrami coefficient of g .If f P W , p C q satisfies (1.1) in C with f p z q ´ z “ O p { z q , then we say that f is the µ -quasiconformal principal mapping. The so-called measurable Riemann mapping theorem (see[AIM09], for instance) grants the existence and uniqueness of the µ -quasiconformal principal map-ping. Moreover, every principal mapping is a homeomorphism of the complex plane.Consider the Sobolev space W s,p p Ω q of functions that are in L p along with all their weakderivatives of order smaller or equal than s P N . Then we can consider the space of traces F s ´ p p,p pB Ω q to be the quotient space W s,p p Ω q{ W s,p p Ω q . Theorem 1.1.
Let s P N and p ą , let Ω be a simply connected, bounded F s ` ´ p p,p -domainand let g : Ω Ñ Ω be a µ -quasiconformal mapping, with supp p µ q Ă Ω and µ P W s,p p Ω q . Then g P W s ` ,p p Ω q . The reader will find the precise definitions in Sections 2 and 3. This theorem also holds forhomeomorphisms between two different simply connected, bounded F s ` ´ p p,p -domains, see Theorem3.5. ∗ KA (Department of Mathematics and Systems Analysis, Aalto University, Finland): [email protected] † MP (Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, Catalonia): [email protected] ‡ ES (Matematiikan ja tilastotieteen laitos, Helsingin Yliopisto, Finland): [email protected] a r X i v : . [ m a t h . A P ] J a n ϕ ϕ f h Ω f p Ω q Figure 1.1: Stoilow decomposition and associated Riemann mappings. Here we assume g : Ω Ñ Ω .We show Theorem 1.1 in Section 3. The idea is to use Stoilow factorization: every µ -quasiconfor-mal mapping g : Ω Ñ Ω can be written as h ˝ f where f is the µ -quasiconformal principal mappingand h : f p Ω q Ñ Ω is conformal (see Figure 1.1). Now, the first step is to show a particular caseof the theorem for f instead of g , which was done in [Pra15b]. Then it remains to show the samefor h . To do so first we need to understand the nature of f p Ω q . Heuristically speaking, the unitnormal vector ν Ω to the boundary of Ω and the unit normal vector ν f p Ω q to the boundary of f p Ω q are related by the formula ν f p Ω q « Df | B Ω ν Ω modulo a scalar normalization, where we considerthe exterior trace of Df . The algebra structure of the spaces under consideration suggests that Df P F s ´ p p,p pB Ω q and ν Ω P F s ´ p p,p pB Ω q should imply that ν f p Ω q has the same regularity and, thus, f p Ω q should be a F s ` ´ p p,p -domain as well. Once we establish that, we will have reduced the theoremto the case µ ”
0, which can be solved using Riemann mappings.At the end of the day, what we use along that scheme are some well-known properties of thespaces (see Lemmas 3.1 and 3.2 below) and the following conditions:1.
Trace condition:
If Ω is a bounded F s ` ´ p p,p -domain and f P W s ` ,p p Ω q , then f | B Ω P F s ` ´ p p,p pB Ω q (the spaces are defined by intrinsec norms, see Section 3.1).2. Riemann mapping condition:
If Ω is a simply connected, bounded F s ` ´ p p,p -domain, thenany Riemann mapping ϕ : D Ñ Ω satisfies that ϕ P W s ` ,p p D q .3. Principal solution condition:
If Ω is a bounded F s ` ´ p p,p -domain, and µ P W s,p p Ω q with } µ } L ă p µ q Ă Ω, then the principal solution is in W s ` ,p p Ω q and it is bi-Lipschitz.For the trace condition we need to check that the definitions we use agree with the well-knownresults in the literature, see Section 3.4. The principal solution condition is given in [Pra15b, The-orem 1.1] and [MOV09, Main Theorem]. Thus, we need to show the Riemann mapping condition,2roven in Section 3.3 using the Pommerenke approach of the Kellogg-Warchawski Theorem (see[Pom92, Theorems 3.5, 3.6]).Let us see how this is done. Consider a conformal map ϕ from the unit disk D to a simplyconnected bounded domain Ω. According to the Kellogg-Warchawski theorem, if the boundary ofΩ is a Dini-smooth Jordan curve, that is, there exists a parameterization w mapping T : “ B D to B Ω such that ż ˜ sup | x ´ y |ă t | w p x q ´ w p y q| ¸ dtt ă 8 , then ϕ has a continuous extension to ¯ D , and ϕ has modulus of continuity given by that of w , andthis result extends as well to the H¨older-continuity of the higher order derivatives of the boundaryparameterizations and the Riemann mapping as well, where the moduli of continuity coincide.The Kellogg-Warchawski Theorem has a natural counterpart for Sobolev spaces (there is anatural counterpart as well for Triebel-Lizorkin spaces which can be found at the end of Section5). Theorem 1.2.
Let s P N and p ą . If Ω is a simply connected, bounded F s ` ´ p p,p -domain, thenany Riemann mapping ϕ : D Ñ Ω is bi-Lipschitz and ϕ P W s ` ,p p D q . Theorem 1.1 explains up to which extent Theorem 1.2 depends on the conformal nature of themappings and the smoothness of the domain under consideration: in the framework of quasicon-formal mappings, that is, replacing the condition ¯ B ϕ “ | ¯ B f | ď κ |B f | with κ ă
1, which givesuniform bounds for the angle distortion of the homeomorphic mappings under consideration, weobtain that the Kellog-Warchawski theorem stays true if the Beltrami coefficient is in the rightfunction space.Theorem 1.1 above has natural counterparts in the context of H¨older-continuous functions and“supercritical” Triebel-Lizorkin functions, at least for fractional smoothness parameters between0 and 1:
Theorem 1.3.
Let ă s ă , let Ω be a simply connected, bounded C s ` -domain and let g : Ω Ñ Ω be a µ -quasiconformal mapping, with supp p µ q Ă Ω and µ P C s p Ω q . Then g P C s ` p Ω q . Theorem 1.4.
Let ă s ă , sp ą and ă q ă 8 , let Ω be a simply connected, bounded F s ` ´ p p,p -domain and let g : Ω Ñ Ω be a µ -quasiconformal mapping, with supp p µ q Ă Ω and µ P F sp,q p Ω q . Then g P F s ` p,q p Ω q . Both theorems above follow the sketch of the Sobolev setting, so we do not repeat the outline ofthe proof for this settings. The authors are convinced that the proof in Section 3 in the frameworkof Sobolev spaces is the one that reflects in a more transparent way the nuances of the problem, soin the remaining cases, we just proof the three properties listed above. In these cases, the readermay use the following dictionaries:Sobolev context H¨older context Triebel-Lizorkin context W s,p p C q C s p C q F sp,q p C q W s ` ,p p C q C s ` p C q F s ` p,q p C q F s ` ´ p p,p p R q C s ` p R q F s ` ´ p p,p p R q We prove Theorem 1.3 (that is, Theorem 1.1 in the H¨older context) in Section 4. In that case,the principal mapping condition can be found in [CF12], and the trace condition is well-known andrather trivial. The Riemann mapping condition is the Kellog-Warchawski Theorem itself.3inally, Section 5 is devoted to study the case of Triebel-Lizorkin spaces. In this setting weobtain great improvements. As in the Sobolev context, the trace condition follows from well-knownresults from the literature via routine computations that we provide for the sake of completeness.The Riemann mapping condition needs again a careful look at Pommerenke’s text, which we do inSection 5.3. The main interest in this part comes from showing the principal mapping condition,which was not in the literature up to now, as far as the authors know, and is summarized in thefollowing theorem, and proven in Section 6.
Theorem 1.5.
Let ă s ă , let ă sp ă 8 (see Figure 1.2), let ă q ă 8 , let Ω be a bounded F s ` ´ { pp,p -domain and let µ P F sp,q p Ω q with } µ } L ă and supp p µ q Ă Ω . Then, the principalsolution f to (1.1) is in the space F s ` p,q p Ω q . The proof follows the scheme of Iwaniec for
V M O
Beltrami coefficients and adapted by Cruz,Mateu and Orobitg for the domain-restricted setting. The key idea is to reduce it to three stepsusing a Fredholm theory argument. First, one needs to show that the Beurling transform (see (2.6)for its definition) restricted to Ω, that is B Ω “ χ Ω B p χ Ω ¨q is bounded in F sp,q p Ω q assuming the Besovregularity in the boundary, which we can find in [CT12] (for cases q P t , p u ). Even more, we needsubexponential quantitative bounds for the iterates of the Beurling transform, which we obtainin Theorem 6.10 (in the present case the growth will be polynomial on the number of iterations).Next we need to show the compactness of the commutator r µ, B Ω s , which was studied in [CMO13]already for more regular domains, but the adaptation to our context is straight-forward, see Lemma6.12.The third step is to check the compactness of the Beurling reflection R : “ χ Ω B p χ Ω c B p χ Ω ¨qq .In Proposition 6.14 we show that not only R is compact in F sp,q p Ω q , but it is in fact smoothing inthe following sense: } R f } F sp,q p Ω q À h } f } C h p Ω q for every h ą
0. To verify that this embedding holds we make use of several techniques, whichinclude the approximation of the boundary of the domain by straight lines as Cruz and Tolsaintroduced in [CT12] which allow us to swap the transform of the characteristic function of thedomain at a given point by a sum of beta coefficients introduced by Dorronsoro in [Dor85]. Wealso use a recent expression of the kernel of the reflection obtained in [Pra15b] (see Section 6.5)and the techniques on chains of Whitney cubes introduced in [PT15, PS17].The quest to understand the regularity of quasiconformal mappings has a long history. Thefirst natural question is to what spaces does the principal solution f belong. Iwaniec showedin [Iwa92] that if the coefficient lays in VMO, that is, in the closure of C in BM O , then thegradient’s integrability is in L p for every 1 ă p ă 8 , using the expression ¯ B f “ p I ´ µ B q ´ p µ q .The break-through in this quest, however, was obtained by the first author in [Ast94], in which thedependence of the integrability of the gradient with respect to } µ } was described with no needof any assumption in the regularity of µ , see [AIS01] for the sharpness and the range of exponents p such that I ´ µ B is invertible in L p . For a result on H¨older regularity see [AIM09, Chapter 15],and for Besov and Triebel-Lizorkin regularity, see [CFM `
09, CFR10, CMO13, BCO17, Pra18].Regarding the study of Beltrami coefficients supported on domains, an extra ingredient isnecessary for the principal solution to inherit the regularity of the Beltrami coefficient, that is,the domain must have a certain degree of smoothness. The first result in this field was given in[MOV09], where they showed that the C s H¨older regularity of the Beltrami coefficient is inheritedby the derivatives of the principal solution to (1.1) as long as C ` s ` ε -domains are considered with0 ă s ă s ` ε ă
1, and later in [CMO13] it was shown that something can be said about theSobolev and Besov regularity as well for these domains. Namely, when 0 ă s ă ă ps ă 8 ,if Ω is a C ` s ` ε -domain, } µ } L ă p µ q Ă Ω, then µ P F sp,p p Ω q ùñ f P F s ` p,p p Ω q , (1.2)4nd the same happens in the scale F sp, . Note that for any interval I , C s ` ` ε p I q Ă C s ` ´ p p I q Ă F s ` ´ p p,p p I q Ă C s ` ´ p p I q , (we used the embeddings in [Tri83, Section 2.7] for the last step) that is, the condition in Theorem1.5 is strictly weaker than the condition in [CMO13], and it seems sharp according to [Tol13].Finally, the last result in this setting can be found in [Pra15b], which studies the case of integervalues for s . The author showed that the parameterizations of the boundary of the domain beingin F s ` ´ p p,p is a sufficient condition for (1.2) to hold, that is, the author showed Theorem 1.5 in thesetting s P N , and q “
2. We devote the appendix to checking that the boundary conditions indifferent papers agree.Theorem 1.5 is an important step towards the proof of a conjecture raised in the thesis of thesecond author (see [Pra15a, Conclusions]). For s P R , we write s “ n s ` t s u , with n s P Z and0 ă t s u ď
1. Note that n s is strictly smaller than s , in particular, it is its integer part when s R Z ,but it is s ´ s is integer. We recall the reader that W s,p “ F sp, for s P N . Conjecture 1.6.
Let s P R , let Ω be a bounded F s ` ´ { pp,p -domain for some ă p t s u ă 8 (seeFigure 1.2) and let µ P F sp, p Ω q with } µ } L ă and supp p µ q Ă Ω . Then, the principal solution f to (1.1) is in the space F s ` p, p Ω q . By the Sobolev embedding (combine [Tri83, Section 2.7] with appropriate extension theorems),this restriction in the indices coincides with the case where f P F sp, p Ω q and all its weak derivativesup to order n s are continuous, and therefore, ordinary derivatives. Regarding the regularity of theboundary, the parameterizations under scope will satisfy exactly the same.Figure 1.2: Conjectures on the indices where Theorem 1.5 holds.The case s P N was proven in [Pra15b, Theorem 1.1]. In this paper we combine the techniquesexposed in [CT12], [CMO13], [Pra15b] and [PS17] to settle the case 0 ă s ă
1. Of course, weexpect Theorem 1.4 to be true in the same range of the previous conjecture.
Throughout this paper we will write C for constants which may change from one occurrence to thenext. If we want to make clear in which parameters C depends, we will add them as a subindex.5n the same spirit, when comparing two quantities x and x , we may write x À x instead of x ď Cx , and x À p ,...,p j x for x ď C p ,...,p j x , meaning that the constant depends on all theseparameters.Given 1 ď p ď 8 we write p for its H¨older conjugate, that is p ` p “ A and B , their symmetric difference is A ∆ B : “ p A Y B qzp A X B q and their longdistance is D p A, B q : “ diam p A q ` diam p B q ` dist p A, B q . Given x P R d and r ą
0, we write B p x, r q or B r p x q for the open ball centered at x with radius r and Q p x, r q for the open cube centered at x with sides parallel to the axis and side-length 2 r .Given any cube Q , we write (cid:96) p Q q for its side-length, and rQ will stand for the cube with the samecenter but enlarged by a factor r . We will use the same notation for balls and one dimensionalcubes, that is, intervals.A domain is an open and connected subset of R d different from H . We say that it is simplyconnected if its complement is connected. Definition 2.1.
Let δ, R ą , d ě . We say that a domain Ω Ă R d is a p δ, R q -Lipschitz domain(or just a Lipschitz domain when the constants are not important) if for every point z P B Ω , thereexists a cube Q “ Q p , R q and a Lipschitz function A z : R d ´ Ñ R supported in r´ R, R s d ´ such that } A z } L ď δ and, possibly after a translation that sends z to the origin and a rotation,we have that Q X Ω “ tp x, y q P Q : y ą A z p x qu . We call such a cube window .If d “ we say that Ω Ă R is a Lipschitz domain if Ω is an open interval. Definition 2.2.
Given a domain Ω , we say that a collection of open dyadic cubes W is a Whitneycovering of Ω if the cubes are disjoint, Ω “ Ť Q P W Q , there exists a constant C W such that C W (cid:96) p Q q ď dist p Q, B Ω q ď C W (cid:96) p Q q , and the family t Q u Q P W has a finite superposition property. Moreover, we will assume that S Ă Q ùñ (cid:96) p S q ě (cid:96) p Q q . The existence of such a covering is granted for any open set different from R d and in particularfor any domain as long as C W is big enough (see [Ste70, Chapter 1] for instance) and may beincreased if needed for our purposes by dividing each cube into its dyadic sons, for instance.The following lemma is true for every Whitney covering. Lemma 2.3 (See [PT15, Lemma 3.11]) . Let d ě . Assume that r ą . If η ą , for every Q P W we have ÿ S P W (cid:96) p S q d D p Q, S q d ` η À (cid:96) p Q q η , (2.1) and ÿ S :D p Q,S qă r (cid:96) p S q d D p Q, S q d ´ η À r η . (2.2) Definition 2.4. If Ω is a Lipschitz domain, for every Q, S P W , we can find a chain r Q, S s , thatis, a sequence of cubes p Q , ¨ ¨ ¨ , Q N q with Q “ Q , Q N “ S , and a central cube Q S : “ Q j for j ď N such that the following holds:If j ď j , then (cid:96) p Q j q « D p Q, Q j q , while (cid:96) p Q j q « D p Q j , S q otherwise, (2.3)6 nd N ÿ j “ (cid:96) p Q j q À D p Q, S q « (cid:96) p Q S q . (2.4)The constants involved depend on the Whitney constants and the Lipschitz character of thedomain. The interested reader may find more information in [PT15, Section 3]. In that paper oneshows that the number of cubes in a chain of a given side-length is uniformly bounded, that is t P P r
Q, S s : (cid:96) p P q “ (cid:96) u ă C. (2.5)More generally, a uniform domain is a domain having a Whitney covering such that for everypair of cubes there exists a chain satisfying (2.3) and (2.4). Moreover, as a consequence it alsosatisfies (2.5) (see [PS17]).When dealing with line integrals in the complex plane, we will write dz for the form dx ` i dy and analogously d ¯ z “ dx ´ i dy , where z “ x ` i y . When integrating a function with respect to theLebesgue measure of a complex variable z we will always use dm p z q to avoid confusion, or simply dm .For any measurable set A and any measurable function f , f A “ ffl A f dm is the mean of f in A .The natural numbers are denoted by N if 0 is not included, and N “ N Y t u . The multiindexnotation for exponents and derivatives will be used: for α P Z its modulus is | α | “ | α | ` | α | andits factorial is α ! “ p α ! qp α ! q . Given two multiindices α, γ P Z we write α ď γ if α i ď γ i forevery i . We say α ă γ if, in addition, α ‰ γ . For z P C and α P Z we write z α : “ z α ¯ z α .We adopt the traditional Wirtinger notation for derivatives, that is, given any φ P C c p C q , then B φ p z q : “ B φ B z p z q “ pB x φ ´ i B y φ qp z q , and ¯ B φ p z q : “ B φ B ¯ z p z q “ pB x φ ` i B y φ qp z q . Given any φ P C c (infintitely many times differentiable with compact support in C ) and α P N wewrite D α φ “ B α φ ¯ B α φ . This notion extends naturally to the tempered distributions (see [Gra08,Chapter 2]).The principal solution to (1.1) can be found using the Beurling transform B ϕ p z q “ lim ε Ñ π ż | w ´ z |ą ε ϕ p w qp z ´ w q dm p w q (2.6)for every ϕ P S . It extends to a bounded operator in L p for every 1 ă p ă 8 , with } B } L Ñ L “ I ´ µ B is invertible in L , and p I ´ µ B q ´ p µ q is a well-definedcompactly supported L function. The principal solution has this function as the antiholomorphicWirtinger derivative, i.e. ¯ B f “ p I ´ µ B q ´ p µ q . Thus, f “ z ` C rp I ´ µ B q ´ p µ qsp z q , where theCauchy transform C is defined as C ϕ p z q “ ´ π ż C ϕ p w q z ´ w dm p w q for every ϕ P S . This expression makes sense for p I ´ µ B q ´ p µ q because it has compact supportand is L p integrable for some p ą
2, see [AIM09, Sections 5.1-5.3].We will write B Ω “ χ Ω B p χ Ω ¨q and B Ω , Ω c “ χ Ω c B p χ Ω ¨q , and similarly for the Cauchy transformor any other operator acting on functions defined in C .Next we define the homogeneous H¨older-Zygmund seminorm:7 efinition 2.5. Given an open set U Ă R d , and ă s ă , we say that f P C s p U q if } f } C s p U q : “ sup x,y P U | f p x q ´ f p y q|| x ´ y | s ă 8 . For s “ we substitute | f p x q´ f p y q| by | f p x q´ f p x ` y q` f p y q| and take the supremum for x, y P U such that this expression makes sense. For k P N and k ă s ď k ` , we say that f P C s p U q if ∇ k f : “ pB k f, B k ´ B f, ¨ ¨ ¨ , B kd f q (that is, a vector with all the partial derivatives of order k ) is in C s ´ k p U q , with } f } C s p U q : “ ›› ∇ k f ›› C s ´ k p U q . Note that the classical homogeneous C functions are Lipschitz, and the latter functions are C . One can define Banach spaces of functions modulo polynomials using the previous seminorms.However, the standard non-homogeneous H¨older-Zygmund spaces are more suitable for our pur-poses: Definition 2.6.
For ă s ă 8 , we say that f P C s p U q if f P L X C s p U q . We define the norm } f } C s p U q : “ } f } L p U q ` } f } C s p U q . The classical Sobolev spaces are defined analogously:
Definition 2.7.
Given s P N and ď p ď 8 , we say that an L loc p U q function f P W s,p p U q if } f } W s,p p U q : “ } ∇ s f } L p p U q ă 8 , where the derivatives are understood in the distributional sense in U . If, moreover, f P L p p U q ,then we write f P W s,p p U q and we define } f } W s,p p U q “ } f } L p p U q ` } f } W s,p p U q . We say that f P W s,ploc p U q if f P W s,p p V q for every open set V contained in a compact subset of U . These intrinsic definitions are not always possible. Thus, one introduces the following:
Definition 2.8.
Let X be a Banach space of complex-valued functions in R d . Given an open set U Ă R d , we say that a measurable function f : U Ñ C belongs to X p U q if } f } X p U q : “ inf F | U ” f } F } X ă 8 . In regular situations, intrinsec definitions will coincide with this general setting. For instance,if Ω is a uniform domain then } f } W s,p p Ω q « inf F | Ω ” f } F } W s,p ă 8 for every s P N , 1 ă p ă 8 by[Jon81], see [Shv10, KRZ15] for extension theorems on worse domains. Definition 2.9.
Let ă s ă 8 , and let Ω be a bounded planar domain. We say that Ω is a C ` s -domain if B Ω is the union of disjoint Jordan curves in a finite collection t Γ j u Mj “ and thereexists a collection of bi-Lipschitz mappings t γ j : B D Ñ Γ j u Mj “ with γ j P C ` s pB D q after the usualidentification of B D with T : “ R {p π Z q . Remark 2.10.
Every C ` s -domain is a Lipschitz domain with a convenient choice of the constants,which depends on the norms of the parameterizations and on the minimum distance between theJordan curves. This can be seen using the Fundamental Theorem of Calculus and the absolutecontinuity of γ j together with the bound below for γ j given by the bi-Lipschitz character.
8o end this introduction we give the definition of Besov and Triebel-Lizorkin spaces. For acomplete treatment we refer the reader to [Tri83].Consider a family t ψ j u j “ Ă C c p R d q satisfying that ř j “ ψ j ” ψ Ă B p , q ,supp ψ j Ă B p , j ` qz B p , j ´ q for j ě
1, and such that for all (cid:126)i P N there exists a constant c (cid:126)i with ››› D (cid:126)i ψ j ››› ď c (cid:126)i j | (cid:126)i | for every j ě . Definition 2.11.
Let F denote the Fourier transform.Let s P R , ď p ď 8 , ď q ď 8 . For any tempered distribution f P S p R d q we define thenon-homogeneous Besov norm } f } B sp,q “ ››(cid:32) sj F ´ ψ j F f (›› (cid:96) q p L p q “ ››(cid:32) sj ›› F ´ ψ j F f ›› L p (›› (cid:96) q , and we call B sp,q Ă S to the set of tempered distributions such that this norm is finite.With the further restriction p ă 8 , we define the non-homogeneous Triebel-Lizorkin norm } f } F sp,q “ ››(cid:32) sj F ´ ψ j F f (›› L p p (cid:96) q q “ ››››(cid:32) sj F ´ ψ j F f p¨q (›› (cid:96) q ›› L p , and we call F sp,q Ă S to the set of tempered distributions such that this norm is finite. These norms are equivalent for different choices of t ψ j u j . Proposition 2.12 (See [Tri83, Sections 2.3.3, 2.5.7 and 2.7.1]) . The following properties hold:1. Given ď p ď p ď 8 and ´8 ă s ď s ă 8 . Then F s p ,p Ă F s p ,p if s ´ p ě s ´ p .
2. Given s ą , then B s , “ C s , in the sense of equivalent norms. Abusing notation we will occasionally write F s , : “ B s , “ C s . In this section we prove Theorem 1.1, that is, the quasiconformal Kellog-Warchawski Theorem forSobolev spaces. We begin by listing in Section 3.1 some well-known properties satisfied by theSobolev spaces to be used in the proof. Next we give the outline of the proof in Section 3.2 andfinally we check the Riemann mapping condition, the trace condition and the principal mappingcondition in sections 3.3, 3.4 and 3.5 respectively.
Let s, d P N , and 1 ď p ď 8 . The test functions are included in the classical space, and fromthe Leibniz’ rule (see [Eva98, Section 5.2.3]) we have that the space W s,p p R d q is closed undermultiplication by C c functions, i.e., for ϕ P C c and f P W s,p , } ϕf } W s,p ď C ϕ } f } W s,p . (3.1)9y the Sobolev embedding Theorem (see [Eva98, Section 5.6]), whenever sp ą d there is a contin-uous embedding into the bounded continuous functions space } f } L ď C } f } W s,p for every f P W s,p . (3.2)Finally, note that if d ě
2, Lipschitz domains are extension domains for W s,p and W s ` ,p (see[Ste70, Theorem VI.5]). The case d “ W s,p p Ω q comes as a consequence of the existence ofa bounded extension operator (see [Tri78, Theorem 4.2.4], for instance). This implies in particularthat } f } W s,p p Ω q « ř | α |ď k } D α f } L p p Ω q by the lifting property (see [Tri83, Theorems 2.3.8, 2.5.6]),i.e., given a bounded Lipschitz domain Ω Ă R d , every function f P W s ` ,p p Ω q satisfies that } f } W s ` ,p p Ω q « } f } W s,p p Ω q ` } ∇ f } W s,p p Ω q . (3.3)Next we state two basic but slightly more specific properties as a separate lemma, which wewill prove in the appendix. Lemma 3.1.
Let s, d P N , and ă p ď 8 . Given bounded Lipschitz domains Ω j Ă R d andfunctions f j P W s,p p Ω j q X W s ´ , p Ω j q with f p Ω q Ă Ω and f bi-Lipzchitz, then f ˝ f P W s,p p Ω q X W s ´ , p Ω q (3.4) and if f p Ω q “ Ω , then f ´ P W s,p p Ω q X W s ´ , p Ω q . (3.5)Since the traces of Sobolev functions are in Besov spaces, we need to list some properties ofthese spaces as well. It is well known that multiplication by any ϕ P C c p R q satisfies that } ϕf } F σp,p ď C ϕ } f } F σp,p for every f P F σp,p p R q (3.6)(see [Tri83, Theorem 2.8.2]). On the other hand, the inequality σp ą } f } L ď } f } F σp,p for every f P F σp,p p R q , (3.7)see Proposition 2.12 above, inducing a multiplicative algebra structure for these spaces.When considering restrictions to intervals, for k P N , k ă σ ă k ` } f } F σp,p p I q : “ ˆż I ż I | f p k q p x q ´ f k p y q| p | x ´ y | p σ ´ k q p ` dy dx ˙ p . (3.8)By [Tri78, Theorem 4.4.2] the norm } f } L p p I q ` } f } F σp,p p I q is equivalent to the one in Definition2.8 whenever 1 ď p ď 8 (see [Tri83, Section 3.4.2] for the endpoints). Moreover, by the liftingproperty [Tri83, Theorem 3.3.5] it holds that } f } F σ ` p,p p I q « } f } F σp,p p I q ` ›› f ›› F σp,p p I q « } f } L p p I q ` ›› f ›› F σp,p p I q . (3.9) Lemma 3.2.
Let k P N , k ă σ ă k ` , ď p ď 8 . Then given bounded intervals I j Ă R ,functions f j P F σp,p p I j q X W k, p I j q with f : I Ñ I being a bi-Lipschitz mapping, then f ˝ f P F σp,p p I q X W k, p I q (3.10) and if f p I q “ I , then f ´ P F σp,p p I q X W k, p I q . (3.11)10he proof is deferred to the appendix as well. Definition 3.3.
Let s ą and sp ą . We say that a Lipschitz domain Ω is a F s ` p,p -domain if B Ω is a finite collection of disjoint Jordan curves t Γ j u Mj “ and there exists a collection of bi-Lipschitzfunctions t γ j : T Ñ Γ j u Mj “ with γ j P F s ` p,p p T q , where T : “ R {p π Z q . Note that, in particular, a F σp,p -domain is a C σ ´ p -domain by Proposition 2.12, and Remark2.10 applies as well. Definition 3.4.
Given σ ą , ă p ă 8 , and given a Jordan curve Γ with a bi-Lipschitzparameterization γ : T Ñ Γ and γ P F σp,p p T q X C p T q , we say that a measurable function f : Γ Ñ C belongs to F σp,p p Γ q if } f } F σp,p p Γ q : “ } f ˝ γ } F σp,p p T q ă 8 . This definition extends naturally to finite collections of Jordan curves.
Note that, since γ P C p T q is bi-Lipschitz, Γ is the boundary of a Lipschitz domain (see Remark2.10 again). In Lemma A.5 we use (3.6) – (3.11) to show that whenever F σp,p Ă C t σ u , the definitionof }¨} F σp,p p Γ q is independent of γ in the sense of equivalent norms. Thus, we could assume withoutloss of generality that the parameterization of the boundary γ is a constant multiple of the arcparameterization (see Lemma A.6). This means that γ is a constant multiple of the rotation of theunit normal vector to the boundary ν at the image point by a right angle, i.e. γ j p z q “ cν p γ p z qq K for z P T . In other words, if k ă s ď k ` p s ´ k q p ą
2, Ω is a F s ` ´ { pp,p -domain exactly when ν P F s ´ { pp,p pB Ω q , condition used in [CT12], [Pra17] (see Proposition A.7). Next we outline the proof of Theorem 1.1. We write the statement in terms of a mapping betweentwo domains, which includes the case of self-maps as a particular case.
Theorem 3.5.
Let s P N and p ą , let Ω and Ω be simply connected, bounded F s ` ´ p p,p -domainsand let g : Ω Ñ Ω be a µ -quasiconformal mapping, with supp p µ q Ă Ω and µ P W s,p p Ω q . Then g P W s ` ,p p Ω q .Proof. By the Stoilow factorization Theorem (see [AIM09, Theorem 5.5.1], for instance) we havethat g “ h ˝ f , where f p z q “ z ` C pp I ´ µ B q ´ p µ qqp z q is the principal solution to (1.1), which is a µ -quasiconformal mapping of the whole complex plane, and h : f p Ω q Ñ Ω is a conformal mapping.We will write Ω : “ f p Ω q . Since f is homeomorphic, we have that Ω is simply connected as well.The first point we need to address is the regularity of Ω . By means of the Riemann mappingtheorem, it is possible to construct a conformal mapping ϕ : D Ñ Ω . By the Kellog-WarchawskiTheorem, ϕ extends to the boundary of D , its derivative extends to the boundary as well and ϕ is bi-Lipschitz in D . By the principal solution condition f is bi-Lipschitz as well (see Remark 3.9).Thus, f ˝ ϕ : T Ñ B Ω is a bi-Lipschitz parameterization (3.12)of the boundary of Ω .Moreover, by the Riemann mapping condition (see Theorem 1.2), we have that ϕ P W s ` ,p p D q . The principal solution condition also grants that f P W s ` ,p p Ω q . By (3.4) the considerationsabove imply f ˝ ϕ P W s ` ,p p D q . Finally, by the trace condition (see Lemma 3.8 below), f ˝ ϕ P F s ` ´ p p,p p T q , (3.13)11nd, combining (3.12) and (3.13), the domain Ω is a F s ` ´ p p,p -domain.Since Ω is simply connected, there exists a Riemann mapping ϕ : D Ñ Ω . This mappingis homeomorphic and holomorphic. To show that g P W s ` ,p p Ω q and it is bi-Lipschitz, it isenough to show that h satisfies both properties and then use the results on f together with (3.4).But h “ p h ˝ ϕ q ˝ ϕ ´ , and both h ˝ ϕ P W s ` ,p p D q and ϕ ´ P W s ` ,p p Ω q by Theorem 1.2again on Ω and Ω respectively, together with (3.5) in the second case. Therefore h inherits bothproperties. For g P L p T q , we define A p z q : “ A p g qp z q “ ż T e it ` ze it ´ z g p t q dt for every z P D . It follows that A p z q “ ż T e it p e it ´ z q g p t q dt. (3.14)Note that ş T e it p e it ´ z q dt “ r i p e it ´ z q ´ s π “ , that is, the kernel in the integral above has meanzero. Thus, we can write A p z q “ ż T e it p g p t q ´ g p s qqp e it ´ z q dt, where z “ re is with 0 ă r ă . (3.15)Next we check that A maps F ´ p p,p p T q into W ,p p D q using the cancellation in (3.15). Howeverwe will give a slightly more general weighted result with the weight δ D p z q : “ ´ | z | for z P D , which will be important in the proof of an analogous result for Triebel-Lizorkin spaces later on. Lemma 3.6.
Let g P L p T q and A : “ A p g q . Given ă σ ď and σ ă p ă 8 , we have that } A } L p p D q ď C } g } L p p T q , and ›› δ ´ σ D A ›› L p p D q ď C } g } F σ ´ pp,p p T q . Proof.
Let us begin with the L p norm. By H¨older’s inequality, } A } pL p p D q “ ż T ż ˇˇˇˇż T e it ` re is e it ´ re is g p t q dt ˇˇˇˇ p rdr ds ď ż T ż ż T | e it ` re is | p | e it ´ re is | p ´ εp | g p t q| p dt ˆż T | e it ´ re is | εp dt ˙ pp rdr ds. Computing the last integral with ε ą p , bounding the first numerator by 2 p and changing theorder of integration, we get } A } L p p D q À ˜ż T | g p t q| p ż T ż p ´ r q pp ´ εp | e it ´ re is | p ´ εp dr ds dt ¸ p . it e is re is ρ Figure 3.1: The “local” part corresponds to the case when re is is inside the grey circle and thedotted segments have comparable lengths.In the last integral above, we distinguish the local and non-local parts (see Figure 3.1). In thenon-local part, when | ´ r | ą | t ´ s | , we have | e it ´ re is | « ´ r . Otherwise, | e it ´ re is | « | t ´ s | .Combining these estimates, and changing ρ : “ ´ r , we can write the last integral as ż | t ´ s | ρ p ´ ´ εp | t ´ s | p ´ εp dρ ` ż | t ´ s | ρ p ´ ´ εp ρ p ´ εp dρ À ` | log p| t ´ s |q| as long as ε ă
1. Since the logarithm is integrable, we obtain } A } L p p D q À ˆż T | g p t q| p ż T p ´ log p| t ´ s |qq ds dt ˙ p À } g } L p , Next we control the norm of the derivative. By (3.15) we have that ›› δ ´ σ D A ›› pL p p D q “ ż T ż ˇˇˇˇż T e it p g p t q ´ g p s qqp e it ´ re is q dt ˇˇˇˇ p p ´ r q p ´ σ q p rdr ds. Using H¨older’s inequality ›› δ ´ σ D A ›› pL p p D q ď ż T ż ż T | g p t q ´ g p s q| p | e it ´ re is | p ´ εp dt ˆż T | e it ´ re is | εp dt ˙ pp p ´ r q p ´ σ q p rdr ds, and choosing again ε P ´ p , ¯ , we get ›› δ ´ σ D A ›› pL p p D q À ż T ż ż T | g p t q ´ g p s q| p | e it ´ re is | p ´ εp dt p ´ r q p ´ ´ εp p ´ r q p ´ σ q p rdr ds. Next we change the order of integration and omit the linear term in r : ›› δ ´ σ D A ›› pL p p D q À ż T ż T | g p t q ´ g p s q| p ˆż p ´ r q p ´ ε ´ σ q p ´ | e it ´ re is | p ´ εp dr ˙ ds dt. Arguing as before, we get ›› δ ´ σ D A ›› pL p p D q À ż T ż T | g p t q ´ g p s q| p ˜ż | t ´ s | ρ p ´ ε ´ σ q p ´ | t ´ s | p ´ εp dρ ` ż | t ´ s | ρ ´ σp ´ dρ ¸ ds dt. ´ σ ´ ε ą
0, which is satisfied whenever ε ă
1, we get ›› δ ´ σ D A ›› L p p D q À ˆż T ż T | g p t q ´ g p s q| p | t ´ s | σp ds dt ˙ p “ } g } F σ ´ pp,p . Corollary 3.7.
The operator A maps F s ´ p p,p p T q to W s,p p D q for s P N and ă p ă 8 .Proof. The case s “ s ě g P F s ´ p p,p p T q . Using(3.14) and integrating by parts, it follows that z p A p g qq p z q “ ż T e it z p e it ´ z q g p t q dt “ ´ i ż T ze it ´ z g p t q dt. Since ş T g “
0, we have that iz p A p g qq p z q “ ż T ˆ ze it ´ z ` ˙ g p t q dt “ ż T e it ` ze it ´ z g p t q dt “ A p g qp z q . Thus, writing G : “ A p g q and G j : “ izG j ´ , the identity above implies G j “ A p g p j q q for 0 ď j ă s. (3.16)Arguing by induction we get A p g q p j q “ z ´ j j ÿ i “ c j,i G i for every j ą . (3.17)Since g p j q P F ´ p p,p p T q for j ă s , from Lemma 3.6 it follows that G j P W ,p p D q by (3.16). Sincethis holds for 0 ď j ď s ´
1, we get that A p g q P W s,p p D z D q by (3.17). This is enough by themaximum principle.To show Theorem 1.2, we will follow the approach of Pommerenke, working with the interplaybetween three elements: first of all, information on the Riemann mapping will be carried by r A p z q “ log ϕ p z q . Secondly, the boundary values of the Riemann mapping will be encoded in afunction γ to be defined below, and finally we will use a parameterization of the boundary of thedomain ω . The smoothness of the latter will then give as a result better regularity of the former. Proof of Theorem 1.2.
By [Pom92, Theorem 3.5] the domain having a C ` ε parameterization im-plies that ϕ is bi-Lipschitz up to the boundary, and this is granted by the embeddings in Proposition2.12 for ε “ ´ p . Therefore, it is sufficient to show that r A P W s,p for r A p z q : “ log ϕ p z q .Let γ p t q : “ arg ϕ p e it q . By the Schwarz integral formula (see [Pom92, Theorem 3.2]) we havethe following relation between the Riemann mapping and the boundary values of its derivative: r A p z q “ i π ż T e it ` ze it ´ z γ p t q dt “ i π A p γ qp z q , (3.18)i.e., r A is the Poisson extension of γ ˝ p e i ¨ q ´ . 14et w : T Ñ B
Ω be a bi-Lipschitz, F s ` ´ p p,p -parameterization. Note that ż T ż T | log p w p s qq ´ log p w p t qq| p | s ´ t | sp ds dt “ ż T ż T | log ´ w p s q w p t q ¯ | p | s ´ t | sp ds dt ď ż T ż T | w p s q w p t q ´ | p | s ´ t | sp ds dt Since w P F s ´ p p,p and | w | «
1, it follows that log w P F s ´ p p,p . Thus,arg w “ Im p log p w qq P F s ´ p p,p (3.19)as well.From [Pom92, Theorem 3.2] again, since Ω has C parameterization of the boundary, we havethat γ p t q “ ´ t ´ π ` r arg w s ˝ “ w ´ ˝ ϕ p e i ¨ q ‰ p t q . (3.20)When s “
1, since ϕ is bi-Lipschitz, the change of variables w ´ ˝ ϕ p e i ¨ q is a bi-Lipschitzhomeomorphism of T . All in all, by (3.8), (3.10), (3.11), (3.19) and (3.20), we have obtained γ P F s ´ p p,p p T q . By (3.18) the theorem follows from Lemma 3.6.If s ą
1, by induction we can assume that r A P W s ´ ,p p D q and ϕ P W s,p p D q . Then, using thefact that traces of the latter space are in F s ´ p p,p pB D q (see [Tri83, Theorem 3.3.3] and Lemma A.8)together with the composition rule (3.10), we get that w ´ ˝ ϕ p e i ¨ q P F s ´ p p,p p T q . We can apply the composition rule again to check that γ P F s ´ p p,p p T q , and r A P W s,p p D q by Corollary3.7 and (3.18), so ϕ P W s ` ,p p D q . Next we show that the trace condition is satisfied. Using Riemann mapping condition we canrecover this result from the classical for C domains. Lemma 3.8.
Let s P N and p ą . If Ω is a F s ` ´ p p,p -domain and f P W s ` ,p p Ω q , then f | B Ω P F s ` ´ p p,p pB Ω q .Proof. First assume that Ω is a simply connected F s ` ´ { pp,p -domain and let f P W s ` ,p p Ω q . Con-sider a Riemann mapping ϕ : D Ñ Ω. By Theorem 1.2, the mapping ϕ is in W s ` ,p p Ω q andbi-Lipschitz. By (3.4) f ˝ ϕ P W s ` ,p p D q as well.Properties (3.1)–(3.5) are used in Lemma A.8 in the appendix to show that the definition ofthe trace space in [Tri83, Definition 3.2.2/2] (via partition of the unity and diffeomorphisms) isequivalent to the one used here by the usual identification of T and B D . In particular, [Tri83, The-orem 3.3] applies and tr p W s ` ,p p D qq “ F s ` ´ { pp,p p T q . Thus, the restriction p f ˝ ϕ q| T P F s ` ´ { pp,p p T q .Since all functions are continuous, the trace is defined pointwise, and p f ˝ ϕ q| T “ f | B Ω ˝ ϕ | T . ByDefinition 3.4, this means that f P F s ` ´ { pp,p pB Ω q .Assume now that Ω is a multiply connected F s ` ´ { pp,p -domain and let f P W s ` ,p p Ω q . Since Ωis a Sobolev extension domain, there is some compactly supported function Ef P W s ` ,p p C q whichcoincides with f in Ω and, therefore, it has the same trace in B Ω.Consider Γ j to be one of the boundary components of B Ω and let Ω j be the bounded simplyconnected domain defined by this Jordan curve, i.e., B Ω j “ Γ j . Let f j : “ Ef | Ω j . Then, as we15ave shown above, f j | B Ω j P F s ` ´ { pp,p pB Ω j q . But f | Γ j coincides with f j | Γ j “ f j | B Ω j , and thus, f | Γ j P F s ` ´ { pp,p p Γ j q . Since this happens with all the components of the boundary of Ω, it followsthat f | B Ω P F s ` ´ p p,p pB Ω q . The principal solution condition is satisfied as well, and the results are already in the literature:
Remark 3.9.
For s P N , ă p ă 8 , if µ is a Beltrami coefficient supported in a bounded F s ` ´ p p,p -domain, then the µ -principal mapping f is bi-Lipschitz and f P W s ` ,p p Ω q .Proof. The bi-Lipschitz character of f comes from [MOV09] because Ω is a C s ` ´ p -domain. Onthe other hand, Proposition A.7 implies that any F s ` ´ p p,p -domain Ω has outward unit normalvector in F s ´ p p,p p Ω q . Thus, we can apply [Pra15b, Theorem 1.1] and f P W s ` ,p p Ω q . In this section we apply the general scheme to prove the H¨older-continuous case.Let 0 ă s ă
1. Note that for d ě C ` s and C s (see [Ste70, Section VI.2.3]). For d “
1, open intervals are extension domains trivially as well.This facts can be used to show that the intrinsic characterization given in Definition 2.6 and therestriction characterization in Definition 2.8 are equivalent norms, i.e. } f } L p Ω q ` sup x,y P Ω | f p x q ´ f p y q|| x ´ y | s « inf F | Ω ” f } F } C s p R d q . and we can use them indistinctly. Also } f } C s ` p Ω q “ } f } L p Ω q ` d ÿ j “ sup x,y P Ω |B j f p x q ´ B j f p y q|| x ´ y | s « inf F | Ω ” f } F } C s ` p R d q . Let 0 ă s ă d P N . As a consequence of the Leibniz’ rule C s p R d q is closed undermultiplication by C c functions, i.e., for ϕ P C c and f P C s , } ϕf } C s ď C ϕ } f } C s . (4.1)By definition there is a continuous embedding into the bounded continuous functions space } f } L ď C } f } C s for every f P C s . (4.2)Given a bounded Lipschitz domain Ω Ă R d , every function f P C s ` p Ω q satisfies that } f } C s ` p Ω q « } f } C s p Ω q ` } ∇ f } C s p Ω q . (4.3)Indeed, to check (4.3) we can use Definition 2.6. Then, we have } f } C ` s p Ω q ď } f } C s p Ω q ` } ∇ f } C s p Ω q trivially, while the converse inequality can be shown using the mean value theorem.16 emma 4.1. Let ă s ă and d P N . Given bounded Lipschitz domains Ω j Ă R d and functions f j P C ` s p Ω j q for j P t , u with f p Ω q Ă Ω and f bi-Lipzchitz, then f ˝ f P C ` s p Ω q and if f p Ω q “ Ω , then f ´ P C ` s p Ω q . See Lemma A.2 in the appendix.We define C s p Γ q as F σ , p Γ q in Definition 3.4. Proposition 4.2.
Let ă s ă . Every bounded C ` s -domain Ω Ă C satisfies the following:1. Trace condition: If f P C ` s p Ω q , then f | B Ω P C ` s pB Ω q .2. Riemann mapping condition: If Ω is simply connected, then any Riemann mapping ϕ : D Ñ Ω satisfies ϕ P C ` s p D q .3. Principal solution condition: If µ P C s p Ω q with } µ } L ă and supp p µ q Ă Ω , then theprincipal solution to (1.1) is in C ` s p Ω q and it is bi-Lipschitz.Proof. The trace condition comes from direct computation. Indeed, let f P C ` s p Ω q (and therefore, f P C ` s p Ω q ), let γ be a C ` s and bi-Lipschitz parameterization of a component of the boundaryof Ω, and let t P T . Then p f ˝ γ q p t q “ Df p γ p t qq γ p t q and, for t , t P R we get |p f ˝ γ q p t q ´ p f ˝ γ q p t q|| t ´ t | s ď | Df p γ p t qq ∇ γ p t q ´ Df p γ p t qq ∇ γ p t q|| t ´ t | s À γ } f } C ` s p Ω q . The last step above can be shown adding and subtracting Df p γ p t qq ∇ γ p t q and using that Df, ∇ γ P L and the bi-Lipschitz character of γ .The Riemann mapping condition comes from [Pom92, Theorem 3.5] and the principal solutioncondition comes from [CF12] and [MOV09, Main Theorem].With this result at hand, Theorem 1.3 follows as in Section 3.2. In this section we proof the quasiconformal Kellogg-Warchawski theorem for Triebel-Lizorkinspaces. In Section 5.1 we recall some results from the literature, including some proofs for thesake of completeness. In Section 5.2 we outline the proof and we check the trace condition andin Section 5.3 we show the Riemann mapping condition. The principal mapping condition is inTheorem 1.5 and its proof is deferred to Section 6.
Let us fix the following notation: Given a domain Ω Ă R d and x P R d , we write δ Ω p x q : “ dist p x, B Ω q . We will use the following characterization for the Triebel-Lizorkin space on Ω:17 heorem 5.1 (see [PS17, Theorem 1.6]) . Let Ω Ă R d be a bounded uniform domain, let ă σ ă , k P N , ă q ď p ă 8 and ă ρ ă . Then f P F sp,q p Ω q for s “ k ` σ if and only if } f } W k,p p Ω q ` ÿ | α |“ k ¨˝ż Ω ˜ż B p x,ρδ p x qq | D α f p x q ´ D α f p y q| q | x ´ y | σq ` d dy ¸ pq dx ˛‚ p ă 8 . Furthermore, the left-hand side of the inequality above is equivalent to the norm } f } F sp,q p Ω q . To allow p ă q we need to take into account the whole shadow of a point. Given ρ ą x P Ω, we define the shadow of x as Sh p x q : “ B p x, ρδ Ω p x qq X Ω . Theorem 5.2 (see [PS17, Theorem 1.6]) . Let Ω Ă R d be a bounded uniform domain, let ă σ ă , k P N and ă p, q ă 8 with σ ą dp ´ dq . Then f P F sp,q p Ω q for s “ k ` σ if and only if } f } W k,p p Ω q ` ÿ | α |“ k ¨˝ż Ω ˜ż Sh p x q | D α f p x q ´ D α f p y q| q | x ´ y | σq ` d dy ¸ pq dx ˛‚ p ă 8 . Furthermore, the left-hand side of the inequality above is equivalent to the norm } f } F sp,q p Ω q . The previous result is based on an extension operator fit to the intrinsic norms defined above.
Theorem 5.3 (see [PS17, Theorem 1.4]) . Let Ω be a uniform domain and k P N . There exists alinear operator Λ k : W k ` , p Ω q Ñ W k ` , p R d q such that for every ă p, q ă 8 and ă σ ă with σ ą dp ´ dq , then Λ k : F sp,q p Ω q Ñ F sp,q p R d q (with s “ σ ` k ) is a bounded operator. The reader may find more general extension operators in the literature. Using extension op-erators, the embeddings described in Proposition 2.12 have the following analogs in the presentsetting:
Proposition 5.4 (See [Tri83, Sections 2.3 and 2.7]) . The following properties hold whenever Ω has an extension operator for the smaller space:1. Let ă q ă q ď 8 and ă p ă 8 and s P R . Then F sp,q p Ω q Ă F sp,q p Ω q .
2. Let ă q , q ď 8 and ă p ď 8 , s P R and ε ą . Then F s ` εp,q p Ω q Ă F sp,q p Ω q .
3. Given ε ą , ă p ă p ă 8 , ă q , q ď 8 and ´8 ă s ă s ă 8 with s P N and s ´ dp “ s ´ dp , then F s p ,q p Ω q Ă F s p ,q p Ω q . To end this introduction, we check the algebra structure of the supercritical spaces, which wewill use in Section 6. 18 emma 5.5.
Let d P N , ă s ă and ds ă p ă 8 . If Ω Ă R d is a Lipschitz domain, then forevery pair f, g P F sp,q p Ω q we have that } f g } F sp,q p Ω q ď C d,s,p,q, Ω } f } F sp,q p Ω q } g } F sp,q p Ω q . (5.1) Moreover, for m P N we have that } f m } F sp,q p Ω q ď C d,s,p,q, Ω m } f } m ´ L p Ω q } f } F sp,q p Ω q . (5.2) Proof.
In [RS96, Theorem 4.6.4/1] it is shown that F sp,q p R d q is a multiplicative algebra. By Theo-rem 5.3, } f g } F sp,q p Ω q ď } Λ f Λ g } F sp,q p R d q ď C } Λ f } F sp,q p R d q } Λ g } F sp,q p R d q ď C d,s,p,q, Ω } f } F sp,q p Ω q } g } F sp,q p Ω q . Regarding inequality (5.2), note that the L p part is trivial. The homogeneous part can bebounded as follows: | f m p x q ´ f m p y q| “ | f p x q ´ f p y q| ˇˇˇˇˇ m ´ ÿ j “ f j p x q f m ´ ´ j p y q ˇˇˇˇˇ ď m } f } m ´ L | f p x q ´ f p y q| . Estimate (5.2) follows from Theorems 5.1 and 5.2.By [Tri83, Theorem 2.8.2] the stability of H¨older spaces under multiplication by smooth func-tions (4.1) has a natural counterpart in the Triebel-Lizorkin scale. Also the embedding propertydescribed in (4.2) has a counterpart whenever sp ą d by [Tri83, Theorem 2.7.1]. Lemma 5.6.
Let ă σ ă , k P N , s “ k ` σ , ă p, q ă 8 and d P N . Then the following holds:1. If σ ą dp ´ dq , given a bounded Lipschitz domain Ω Ă R d , every function f P F s ` p,q p Ω q satisfiesthat } f } F s ` p,q p Ω q « } f } F sp,q p Ω q ` } ∇ f } F sp,q p Ω q . (5.3)
2. If σp ą d , given bounded Lipschitz domains Ω j Ă R d and functions f j P F s ` p,q p Ω j q with f p Ω q Ă Ω and f bi-Lipzchitz, then f ˝ f P F s ` p,q p Ω q (5.4) and if f p Ω q “ Ω , then f ´ P F s ` p,q p Ω q . (5.5) Proof.
The lifting property (5.3) is a straightforward consequence of Theorems 5.1 and 5.2. Theproofs of (5.4) and (5.5) are given in Lemma A.4 at the appendix.
Now we concentrate again on the two-dimensional case.
Theorem 5.7.
Let ă s ă , ă sp ă 8 , ă q ă 8 . Every bounded F s ` ´ p p,p -domain Ω Ă C satisfies the following:1. Trace condition: If f P F s ` p,q p Ω q , then f | B Ω P F s ` ´ p p,p pB Ω q .2. Riemann mapping condition: If Ω is simply connected, then any Riemann mapping ϕ : D Ñ Ω satisfies that ϕ P F s ` p,q p D q . NEW PAGE . Principal solution condition: If µ P F sp,q p Ω q with } µ } L ă and supp p µ q Ă Ω , then theprincipal solution to (1.1) is in F s ` p,q p Ω q and it is bi-Lipschitz.Proof. The Riemann mapping condition is in Theorem 5.11 below. The principal mapping condi-tion is in Theorem 1.5 and will be shown in Section 6. In turn, the trace condition is a consequenceof the following lemma.
Lemma 5.8.
Let ă σ ă , k P N , s “ k ` σ and ă p, q ă 8 with σp ą . If Ω is a F s ` ´ p p,p -domain and f P F s ` p,q p Ω q , then f | B Ω P F s ` ´ p p,p pB Ω q .Proof. One applies the proof of Lemma 3.8, using Lemma 5.6 instead of Lemma 3.1 and using theRiemann mapping condition in Theorem 5.11 instead of Theorem 1.2
Lemma 5.9.
For every analytic function F , ă σ ă and ρ ă , ă q ď p ă 8 , we have that } F } F σp,q p D q : “ ¨˝ż D ˜ż B p z,ρδ D p z qq | F p z q ´ F p ξ q| q | z ´ ξ | σq ` dξ ¸ pq dz ˛‚ p À ›› δ ´ σ F ›› L p p D q . Proof.
We use the mean value property of analytic functions: given z P D and ξ P B p z, ρδ D p z qq ,we have that | F p z q ´ F p ξ q| q | z ´ ξ | q ď ›› F ›› qL pr z,ξ sq ď C q ˜ B p z, ρδ D p z qq | F p ζ q| dm p ζ q ¸ q . Computing and using the Jensen inequality we obtain } F } p F sp,q p D q À ż D ˜ż B p z,ρδ D p z qq | z ´ ξ | p σ ´ q q ` dξ B p z, ρδ D p z qq | F p ζ q| q dζ ¸ pq dz À ρ ż D ˜ δ D p z q p ´ σ q q B p z, ρδ D p z qq | F p ζ q| q dζ ¸ pq dz ď ż D δ D p z q p ´ σ q p B p z, ρδ D p z qq | F p ζ q| p dζ dz To conclude, note that δ D p z q « δ D p ζ q in the integral above, so } F } p F sp,q p D q À ρ ż D δ D p ζ q p ´ σ q p | F p ζ q| p B p ζ, ρδ D p z qq dz dζ “ ›› δ ´ σ F ›› L p p D q . Proposition 5.10.
Let s “ k ` σ with k P N , ă σ ă . The operator A maps F s ´ p p,p p T q to F sp,q p D q for σ ă p ă 8 and ă q ă 8 .Proof. First note that it is enough to show the case p ě q , since otherwise F sp,p p D q Ă F sp,q p D q . Let g P F s ´ p p,p p T q and consider G j defined as in the proof of Corollary 3.7. By the lifting property for20mooth domains (see [Tri83, Theorem 3.3.5]) and Theorem 5.1 we have } A p g q} F sp,q p D q « } A p g q} W k,p p D q ` ¨˝ż D ˜ż B p x,ρδ D p x qq | A p g q p k q p x q ´ A p g q p k q p y q| q | x ´ y | σq ` d dy ¸ pq dx ˛‚ p (5.6) « } A p g q} W k,p p D q ` ¨˝ż D z D ˜ż B p x,ρδ D p x qq | A p g q p k q p x q ´ A p g q p k q p y q| q | x ´ y | σq ` d dy ¸ pq dx ˛‚ p , where we have used Caccioppoli and Harnack inequalities to omit the central region in the lastintegral. By (3.16), Lemma 5.9 and Lemma 3.6, it follows that for j ď k , } G j } F σp,q p D q “ ››› A p g p j q q ››› F σp,q p D q À ››› δ ´ σ A p g p j q q ››› L p p D q À ››› g p j q ››› F σp,p p T q , and an analogous result can be obtained for the non-homogeneous part. Using (3.17) and (5.6),the proposition follows. Theorem 5.11.
Let s “ k ` σ , with k P N a natural number and ă σ ă , ă p ă 8 with σp ą and ă q ď 8 . If Ω is a simply connected F s ` ´ p p,p -domain, then any Riemann mapping ϕ : D Ñ Ω is bi-Lipschitz and ϕ P F s ` p,q p D q .Proof. The proof runs parallel to that of Theorem 1.2. Recall that we only need to show that r A P F sp,q p D q for r A p z q : “ log ϕ p z q . Let γ p t q : “ arg ϕ p e it q . Let w : T Ñ B
Ω be a (locally)bi-Lipschitz, F s ` ´ p p,p -parameterization.We have seen that r A p z q “ i π A p γ qp z q , and arg w “ Im p log p w qq P F s ´ p p,p as well, whereas both parameterizations are related by γ p t q “ ´ t ´ π ` r arg w s ˝ “ w ´ ˝ ϕ p e i ¨ q ‰ p t q . When s ă
1, since ϕ is bi-Lipschitz, the change of variables w ´ ˝ ϕ p e i ¨ q is a bi-Lipschitzhomeomorphism of T . Using (3.8), (3.10) and (3.11), we obtain that γ P F s ´ p p,p p T q . By Proposition5.10, we get that r A P F sp,q p D q , so ϕ P F s ` p,q p D q and the theorem follows.If s ą
1, by induction we can assume that r A P F s ´ p,q p D q and ϕ P F sp,q p D q . Then, using the factthat traces of the latter space are in F s ´ p p,p pB D q together with the composition rule (3.10) and theinverse function rule (3.11), we get that w ´ ˝ ϕ p e i ¨ q P F s ´ p p,p p T q . We can apply the composition rule again to check that γ P F s ´ p p,p p T q , and the theorem follows asbefore from Proposition 5.10. 21 Proof of Theorem 1.5
In this section we prove Theorem 1.5. In Section 6.1 we outline its proof, which follows thesteps of [Pra15b] by means of a classical Fredholm argument, reducing the proof to checkingthat I Ω ´ µ m p B m q Ω is invertible, and that the commutator r µ, B Ω s and the Beurling reflection χ Ω B p χ Ω c B p χ Ω ¨qq is compact (together with a family of related operators).After that we recall Dorronsoro’s Betas in Section 6.2, which will be our tool to measure theflatness of the boundary in a multi-scale basis. Next we show that the iterates of the truncatedBeurling transform are bounded with subexponential growth (polynomial in fact) in Section 6.3,which will allow us to find the invertible part of the Fredholm operator. We check the compactnessof the commutator in Section 6.4. Finally we prove the compactness of the Beurling Reflectionin Section 6.5 in what represents the most difficult challenge in this paper, leaving a technicallemma to be shown in Section 6.6 where Meyer’s polynomials are introduced to control oscillationin Whitney cubes. First we will face the invertibility of p I ´ µ B qp χ Ω ¨q in F sp,q p Ω q . Here χ Ω g denotes the extensionof a given function g P F sp,q p Ω q by zero in Ω c . We will follow the scheme used in [Iwa92]. Thatis, we will reduce the proof to the compactness of the commutator. In our context, however, asit happens in [CMO13] and [Pra15b], we will have to deal with the compactness of operators like χ Ω B p χ Ω c B p χ Ω ¨qq as well.Consider m P N . Recall that p B m q Ω g “ χ Ω B m p χ Ω g q for g P L p p Ω q and I Ω be the identityon F sp,q p Ω q . Let us define P m : “ I Ω ` µ B Ω ` p µ B Ω q ` ¨ ¨ ¨ ` p µ B Ω q m ´ . We will check thatthe truncated Beurling transform is bounded on F sp,q p Ω q in Theorem 6.10 below. Since F sp,q p Ω q is a multiplicative algebra (under the conditions of Lemma 5.5), we have that P m is bounded in F sp,q p Ω q . Note that P m ˝ p I Ω ´ µ B Ω q “ p I Ω ´ µ B Ω q ˝ P m “ I Ω ´ p µ B Ω q m , (6.1)and I Ω ´ p µ B Ω q m “ p I Ω ´ µ m p B m q Ω q ` µ m pp B m q Ω ´ p B Ω q m q ` p µ m p B Ω q m ´ p µ B Ω q m q“ A p q m ` µ m A p q m ` A p q m . (6.2)Note the difference between p B Ω q m g “ χ Ω B p . . . χ Ω B p χ Ω B p χ Ω g qqq and p B m q Ω g “ χ Ω B m p χ Ω g q .We want to check that for m large enough, the operator I Ω ´ p µ B Ω q m is the sum of an invertibleoperator and a compact one.First we will study the compactness of A p q m “ µ m p B Ω q m ´p µ B Ω q m . To start, writing r µ, B Ω sp¨q for the commutator µ B Ω p¨q ´ B Ω p µ ¨q we have the telescopic sum A p q m “ m ´ ÿ j “ µ m ´ j r µ, B Ω s ` µ j ´ p B Ω q m ´ ˘ ` p µ B Ω q A p q m ´ . Arguing by induction we can see that A p q m can be expressed as a sum of operators bounded in F sp,q p Ω q which have r µ, B Ω s as a factor. It is well-known that the compactness of a factor impliesthe compactness of the operator (see for instance [Sch02, Section 4.3]). Below, in Lemma 6.12 weverify that the commutator r µ, B Ω s is compact in F sp,q p Ω q .Consider now A p q m “ p B m q Ω ´ p B Ω q m . We define the operator R m g : “ χ Ω B p χ Ω c B m p χ Ω g qq (6.3)22henever it makes sense. This operator can be understood as a (regularizing) reflection withrespect to the boundary of Ω, and it is bounded in F sp,q p Ω q again by Theorem 6.10 below. Notethat R m ´ “ χ Ω r χ Ω , B s ˝ B m ´ p χ Ω ¨q , leading to A p q m “ R m ´ ` B Ω ˝ A p q m ´ . Thus, the reflectionis bounded and the compactness of R m shown in Theorem 6.13 will prove the compactness of A p q m .Now, the following claim is the remaining ingredient for the proof of Theorem 1.5. Lemma 6.1.
For m large enough, A p q m is invertible.Proof. Since sp ą g P F sp,q p Ω q} µ m p B m q Ω g } F sp,q p Ω q À } µ m } F sp,q p Ω q }p B m q Ω g } F sp,q p Ω q À m } µ } m ´ L } µ } F sp,q p Ω q }p B m q Ω } F sp,q p Ω qÑ F sp,q p Ω q } g } F sp,q p Ω q . By Theorem 6.10 below, there are constants depending on the Lipschitz character of Ω (andother parameters) but not on m , such that }p B m q Ω } F sp,q p Ω qÑ F sp,q p Ω q À m } ν } F s ´ { pp,p pB Ω q . As a consequence, for m large enough the operator norm } µ m p B m q Ω } F sp,q p Ω qÑ F sp,q p Ω q ă A p q m in (6.2) is invertible.Now we can show Theorem 1.5 for 0 ă s ă Proof of Theorem 1.5.
For m big enough, the restricted Beltrami operator I Ω ´ p µ B Ω q m can beexpressed using (6.2) as the sum of an invertible operator A p q m (see Lemma 6.1) and the compactoperator µ m A p q m ` A p q m (its compactness granted in the comments above together with Lemma 6.12and Theorem 6.13). By (6.1), we can deduce that I Ω ´ µ B Ω is a Fredholm operator (see [Sch02,Theorem 5.5]). The same argument works with any other operator I Ω ´ tµ B Ω for 0 ă t ă {} µ } .It is well known that the Fredholm index is continuous with respect to the operator norm onFredholm operators (see [Sch02, Theorem 5.11]), so the index of I Ω ´ µ B Ω equals that of I Ω , i.e.,0. It only remains to see that our operator is injective in order to obtain its invertibility. Sincethe Beurling transform is an isometry on L p C q and } µ } ă
1, the operator I ´ µ B is injective in L p C q . Thus, if g P F sp,q p Ω q , and p I Ω ´ µ B Ω q g “
0, we define G p z q “ g p z q if z P Ω and G p z q “ p I ´ µ B q G p z q “ p I ´ µχ Ω B qp χ Ω G qp z q “ p I Ω ´ µ B Ω q g p z q “ z P Ω0 otherwise . By the injectivity of the first operator, since G P L we get that G “ g “ F sp,q p Ω q .Now, remember that the principal solution of (1.1) is f p z q “ C h p z q ` z , where h : “ p I ´ µ B q ´ µ, that is, h ´ µ B p h q “ µ , so supp p h q Ă supp p µ q Ă Ω and, thus, χ Ω h ` µ B Ω p h q “ h ` µ B p h q “ µ modulo null sets, so h | Ω “ p I Ω ´ µ B Ω q ´ µ, proving that h | Ω P F sp,q p Ω q . By [AIM09, Theorem 4.3.12] we have that C h P L p p C q . Since thederivatives of the principal solution, B f | Ω “ h | Ω and B f | Ω “ B Ω h `
1, are in F sp,q p Ω q , we have f P F s ` p,q p Ω q by (5.3). 23 .2 Dorronsoro’s Betas Jose R. Dorronsoro introduced the following polynomials to study the Besov norms of functions in[Dor85] (see [Pra17, Proposition 2.3] for the consistency of this definition):
Definition 6.2.
Let I be an interval and let f P L p I q . Then, there exists a unique polynomial R nI f of degree n (or smaller) such that for every j P t , , ¨ ¨ ¨ , n u , ż I p R nI f ´ f q x j dm “ . Then, we define β p n q p f, I q : “ (cid:96) p I q ż I | f p x q ´ R nI f p x q| (cid:96) p I q dm p x q . The β -coefficients are closely related to Jones-David-Semmes ones. Namely, if f is Lipschitzand n “
1, then β p q « β . On the other hand, these polynomials satisfy that } R nI f } L p I q À n (cid:96) p I q ´ } f } L p I q . (6.4)Let D d stand for a dyadic grid of R d . As it was observed in [Pra17, (2.10)], one can rewrite[Dor85, Theorem 1] in terms of these coefficients as follows. Lemma 6.3.
Let ă s ă n ` and ď p ă 8 . Then for every f P F sp,p p R q , we have that } f } B sp,p « ˜ ÿ I P D ˆ β p n q p f, I q (cid:96) p I q s ´ ˙ p (cid:96) p I q ¸ p . In the case p “ 8 , arguing analogously, we get } f } C s « } f } B s , « sup I P D β p n q p f, I q (cid:96) p I q s ´ . We will use the beta coefficients to measure the regularity of a domain. Namely, we measure inevery scale how far is each portion of the boundary to be the graph of a polynomial, via a dyadicapproach. To make the notation less dense, we will assign the coefficients to the Whitney cubesstraight ahead. To do so, we will chose a beta coefficient comparable to the supremum of thebetas of the reasonable choices for each cube. In the following definitions and computation we use ε δ : “ R ? ` δ , which grants that whenever x ă ε δ , the image p x, A p x qq under a parameterization A is a boundary point even if the Lipschitz constant δ is big. Definition 6.4.
Let Ω Ă C be a bounded p δ, R q -Lipschitz domain. Take a finite collection ofboundary points t x j u Mj “ Ă B Ω such that (cid:32) B ` x j , ε δ ˘( Mj “ is a disjoint family but the double ballscover the boundary. After an appropriate rigid movement τ j (rotation and translation) which maps x j to the origin, the boundary Bp τ j Ω q coincides with the graph of a Lipschitz function A j in thecube Q j “ Q p , R q , with A j supported in r´ R, R s and derivative satisfying } A } L ď δ .Given a cube Q P W Ω with (cid:96) p Q q small enough, say (cid:96) p Q q ď C Ω , we say that a pair p x j , J q isadmissible for Q , writing p x j , J q P J Q if1. The length (cid:96) p J q “ (cid:96) p Q q and J Ă r´ ε δ , ε δ s .2. The image of J under the graph function τ ´ j ˝ p Id, A j q is a set U J Ă B Ω with D p Q, U J q « (cid:96) p Q q . bove we can fix the constants so that every pair p x j , J q with (cid:96) p J q ď C Ω and J Ă r´ ε δ , ε δ s belongsat least to one J Q . Then, we assign the number β p n q p Q q : “ max p x j ,J qP J Q β p n q p A j , J q . If the cube is greater than C Ω , we will assign β p n q p Q q : “ . Remark 6.5.
Note that the number of candidates J above is uniformly bounded in terms of theLipschitz character and the Whitney constants. At the same time, every interval J can be chosenfor a uniformly bounded number of Whitney cubes depending on the same constants. Lemma 6.6 (see Proposition A.9) . Let ă s ă n , ă p ă 8 with sp ą and let Ω be a p δ, R q -Lipschitz F s ` ´ p p,p -domain. Then ¨˝ ÿ Q P W : (cid:96) p Q qď C Ω β p n q p Q q p (cid:96) p Q q ´ sp ˛‚ p À } ν } F s ´ pp,p pB Ω q , with constants depending on the Lipschitz character of Ω and H pB Ω q , where ν stands for the unitoutward normal vector to the boundary of the domain. The β -coefficients will appear in a natural way along the present paper thanks to the followingrelation introduced in [CT12, (7.3)]. Lemma 6.7.
Let Ω be a bounded p δ, R q -Lipschitz domain and let W be a Whitney covering withappropriate constants. Then, for x, y P Q P W with Q Ă Ť Mj “ B ` x j , ε δ ˘ , there exists a half plane Π Q so that for every ă (cid:96) ă R , the estimate ż Ω∆Π Q | z ´ x | ` η dm p z q À ¨˚˚˝ ÿ P P W : ρP Ą Q(cid:96) p P qď (cid:96) β p q p P q (cid:96) p P q η ` (cid:96) η ˛‹‹‚ (6.5) holds, with the constant ρ ą depending only on the Lipschitz character of the domain and η ą . Note that the condition ρP Ą Q in the last sum above, implies that the cubes P cannot bemuch smaller than Q , namely (cid:96) p P q ě (cid:96) p Q q ρ , and thus the number of cubes P on a given scale staysuniformly bounded for any given Q . Essentially Π Q is a half-plane whose boundary coincides withthe minimizer for β p q p Q q . To be precise, we choose p x j , J q P J Q and we choose Π Q so that itcontains Q and τ j B Π Q minimizes β p n q p A j , J q , see Definition 6.4. Above we chose the Whitneyconstants big enough so that that dist p Q, Π cQ q « (cid:96) p Q q . Note that in case (cid:96) p Q q ě C Ω we can choseany half-plane whose boundary is at distance from Q comparable to (cid:96) p Q q .To end with beta coefficients, we write the following lemma, which will be used several timesalong the text. Lemma 6.8.
Let Ω Ă C be a bounded F s ` ´ p p,p -domain, with ă s ă , ă p ă 8 and sp ą .For (cid:96) ą s , we have that ÿ Q P W (cid:96) p Q q `p (cid:96) ´ s q p ¨˝¨˝ ÿ ρP Ą Q : (cid:96) p P qď R β p q p P q (cid:96) p P q (cid:96) ˛‚ p ` ˛‚ À } ν } pF s ´ pp,p pB Ω q , with constants depending on (cid:96), s, p , the Whitney constants, the Lipschitz character of the domainand its diameter. roof. Fix ε ă (cid:96) ´ s . ThenSS : “ ÿ Q P W (cid:96) p Q q `p (cid:96) ´ s q p ¨˝¨˝ ÿ ρP Ą Q : (cid:96) p P qď R β p q p P q (cid:96) p P q (cid:96) ˛‚ p ` ˛‚ À ˜ C `p (cid:96) ´ s q p Ω ` ÿ Q (cid:96) p Q q `p (cid:96) ´ s q p ÿ ρP Ą Q β p q p P q p (cid:96) p P q p (cid:96) ´ ε q p ˆ (cid:96) p Q q εp ˙ pp ¸ À ˜ C Ω ,(cid:96) ´ s,p ` ÿ P β p q p P q p (cid:96) p P q p (cid:96) ´ ε q p ÿ Q Ă ρP (cid:96) p Q q `p (cid:96) ´ s ´ ε q p ¸ . Now, using Remark 6.5, for P P W , since 2 ` p (cid:96) ´ s ´ ε q p ą ÿ Q Ă ρP (cid:96) p Q q `p (cid:96) ´ s ´ ε q p « ÿ J Ă I : I P J p P q (cid:96) p J q `p (cid:96) ´ s ´ ε q p ď ÿ I P J p P q (cid:96) p I q `p (cid:96) ´ s ´ ε q p « (cid:96) p P q `p (cid:96) ´ s ´ ε q p . Thus, by Lemma 6.6 (see Proposition A.9 in case 1 ă sp ď
2) we getSS À ˜ C Ω ,(cid:96) ´ s,p ` ÿ P β p q p P q p (cid:96) p P q sp ´ ¸ ď C Ω ,(cid:96) ´ s,p } ν } pF s ´ pp,p pB Ω q . Remark 6.9.
Since we are in a Lipschitz domain, it is enough ` p (cid:96) ´ s ´ ε q p ą , see [PT15,Lemma 3.12]. Thus, the preceding lemma holds in fact whenever (cid:96) ą s ´ p . Consider 1 ă p ă 8 , sp ą
2, and 0 ă s ă F s ` ´ p p,p -domain. Victor Cruzand Xavier Tolsa showed that for every f P F sp, p Ω q we have that } B p χ Ω f q} F sp, p Ω q ď C } ν } F s ´ { pp,p pB Ω q } f } F sp, p Ω q , where C depends on p , s , diam p Ω q and the Lipschitz character of the domain. (see [CT12, Corollary1.3]).Nevertheless, this estimate is not enough, since we need to estimate the iterates of the Beurlingtransform, that is, Theorem 6.10 below. Moreover, we are dealing with the larger Triebel-Lizorkinscale (with values other than 2 allowed for q as well). Thus, we proceed to give a quantitativecontrol of }p B m q Ω } F sp,q p Ω q . The following is a fractional version of what we got in [Pra17]. Theorem 6.10.
Consider m P N , ă s ă and ă p, q ă 8 with sp ą , and let Ω be a bounded F s ` ´ p p,p -domain. Then, for every f P F sp,q p Ω q we have that } B m p χ Ω f q} F sp,q p Ω q ď Cm } ν } F s ´ { pp,p pB Ω q } f } F sp,q p Ω q , where C depends on s , p , q , diam p Ω q and the Lipschitz character of the domain but not on m . Before proving that, we check the behavior on constant functions, which will be enough whencombined with the T p q Theorem in [PS17]. 26 emma 6.11.
Consider m P N , let ă s ă , ă p, q ă 8 with sp ą , let Ω be a bounded F s ` ´ p p,p -domain. Then B m χ Ω P F sp,q p Ω q and, moreover, } B m χ Ω } F sp,q p Ω q À } ν } F s ´ pp,p pB Ω q , the constant depending only on the indices, the Lipschitz character of the domain and its diameter.Proof. Note that if p ă q , then } B m χ Ω } F sp,q p Ω q À } B m χ Ω } F sp,p p Ω q , so we will assume with no lossof generality that p ě q , which in particular implies that s ą ě p ´ q . We follow the approachgiven in [CT12, proof of Lemma 6.3] which gets quite shorter with the norm given in Theorem 5.1if p ě q . Indeed, we only need to control the homogeneous seminorm } B m χ Ω } p F sp,q p Ω q : “ ÿ Q P W ż Q ˆż Q | B m χ Ω p x q ´ B m χ Ω p y q| q | x ´ y | sq ` dy ˙ pq dx, (6.6)and the non-local part in the aforementioned proof, which is the most difficult one to treat, is notthere anymore.Choose a half-plane Π Q as in Lemma 6.7. By (6.4), chosing appropriate Whitney constantswe have that x and y are in Π Q . Next we use that, formally, B χ Π Q is constant in Π Q and Π Qc (see [CT12, Lemma 4.2]), i.e., that B χ Π Q “ cχ Π Q modulo constants and, by induction, B m χ Π Q is constant in Π Q as well, so | B m χ Ω p x q ´ B m χ Ω p y q| “ | B m χ Ω p x q ´ B m χ Ω p y q ´ B m χ Π Q p x q ` B m χ Π Q p y q| (understood in the BMO sense).To avoid checking the particular kernel of the iterates of the Beurling transform in BMO, whichwe could not find in the literature, the reader can check that ż r D X Π Q z B p x,ε q p z ´ x q m ´ p z ´ x q m ` dm p z q ´ ż r D X Π Q z B p y,ε q p z ´ y q m ´ p z ´ y q m ` dm p z q r Ñ8 ÝÝÝÑ ż p r D q c p z ´ x q m ´ p z ´ x q m ` ´ p z ´ y q m ´ p z ´ y q m ` dm p z q r Ñ8 ÝÝÝÑ . Thus, we can write | B m χ Ω p x q ´ B m χ Ω p y q| ď m ż Ω∆Π Q ˇˇˇˇ p z ´ x q m ´ p z ´ x q m ` ´ p z ´ y q m ´ p z ´ y q m ` ˇˇˇˇ dm p z qÀ m ż Ω∆Π Q | x ´ y || z ´ x | dm p z q . By (6.5), we can write | B m χ Ω p x q ´ B m χ Ω p y q| À m | x ´ y | ˜ ÿ P P W : Q Ă ρP β p q p P q (cid:96) p P q ` p Ω q ¸ . Combining with (6.6), we get that } B m χ Ω } p F sp,q p Ω q À ÿ Q P W | Q | ˜ (cid:96) p Q q p ´ s q q m q ˜ ÿ P : Q Ă ρP β p q p P q (cid:96) p P q ` p Ω q ¸ q ¸ pq « m p ÿ Q P W (cid:96) p Q q ` p ´ sp ˜ ÿ P : Q Ă ρP β p q p P q (cid:96) p P q ` p Ω q ¸ p and Lemma 6.8 with (cid:96) “ roof of Theorem 6.10. By [PS17, Remark 5.7], we have that } B Ω } F sp,q p Ω qÑ F sp,q p Ω q À s,p,q, diam p Ω q ,δ m ` } B } F sp,q Ñ F sp,q ` } B } L p Ñ L p ` } B } L q Ñ L q ` } B Ω } F sp,q p Ω q The boundedness of the Beurling transform in F sp,q (see [Tor91]) and Lemma 6.11 imply that } B Ω } F sp,q p Ω qÑ F sp,q p Ω q À s,p,q, diam p Ω q ,δ m ´ } ν } F s ´ { pp,p pB Ω q ` ¯ « s,p,q, diam p Ω q ,δ m } ν } F s ´ { pp,p pB Ω q . Lemma 6.12.
The commutator r µ, B Ω s is compact in F sp,q p Ω q .Proof. Choose s ă β ă
1. If µ P C p C q , then r µ, B Ω s : L Ñ W β,p is compact (see [CMO13, (18)and the subsequent paragraph]). Precomposing with the inclusion F sp,q Ñ L and postcomposingwith W β,p Ñ F sp,q for β ą s we get that the lemma holds for C coefficients.To show compactness for general µ P F sp,q p Ω q , we only need to see that the commutator can beapproximated in operator norm by a sequence of commutators with smooth coefficients. For thisone only needs to approximate µ by t µ n u Ă C p ¯Ω q (combine the density of C c functions in F sp,q in [Tri83, Theorem 2.3.3] and Theorem 5.3, for instance). By (5.1) and Theorem 6.10 ([CT12] isenough in this case), we conclude that r µ n , B Ω s Ñ r µ, B Ω s in the operator norm. Theorem 6.13.
Let ă s ă , ă p, q ă 8 with sp ą , and let Ω be a bounded F s ` ´ p p,p -domain.For every m , the operator R m is compact in F sp,q p Ω q . Theorem 6.13 is a straight consequence of the Rellich-Kondrachov compactness Theorem (see[Tri83, Remark 4.3.2/1]) together with the following proposition.
Proposition 6.14.
Let ă s ă , ă p, q ă 8 with sp ą , and let Ω be a bounded F s ` ´ p p,p -domain, and m P N . Then } R m f } F sp,q p Ω q À h } f } C h p Ω q for every h ą . Next we take a closer look to the kernel of the Beurling reflection defined in (6.3). The reflectioncan be written as R m f p z q “ ż Ω c p z ´ w q ż Ω f p ξ q p w ´ ξ q m ´ p w ´ ξ q m ` dm p ξ q dm p w q . In a quite general setting, one can use Fubini in the former expression of R m and the relatedkernel r K m p z, ξ q “ ş Ω c p w ´ ξ q m ´ p z ´ w q p w ´ ξ q m ` dw appears as a natural element. Mateu, Orobitg andVerdera study this kernel in [MOV09, Lemma 6] assuming the boundary of the domain Ω to be in C ` ε for ε ă
1. They prove the size inequality | r K m p z, ξ q| À | z ´ ξ | ´ ε C ` s ` ε parameterizations. They could show that the kernel issmoothing in this context. Their proof was based on the size and the smoothness estimates of thekernel shown in [MOV09], which could be useful for the case W σ,p p Ω q with σ ă s ´ { p but they arenot sufficiently strong to deal with the endpoint case W s,p p Ω q when the domain has just F ` s ´ p p,p parameterizations. Nevertheless, their argument was adapted in [Pra15b] to get Proposition 6.18below, which will be used to prove Proposition 6.14.Let us collect the necessary background. Given m P N , let us define the kernel K m p z, ξ q : “ ż Ω c p w ´ ξ q m ´ p z ´ w q p w ´ ξ q m ` dm p w q “ ż B Ω p w ´ ξ q m p z ´ w q p w ´ ξ q m ` dw (6.7)for all z, ξ P Ω, where the path integral is oriented counterclockwise. Note that for suitable z and f we will be able to use Fubini’s Theorem to get B R m f p z q “ ż Ω f p ξ q K m p z, ξ q dm p ξ q . Lemma 6.15.
Let Π be an open half plane, and x, y P Π . For m , m , m P N with m ` m ´ m ą we have that ż Π c p w ´ y q m p x ´ w q m p w ´ y q m dm p w q “ ż B Π p w ´ y q m ` p x ´ w q m p w ´ y q m dw “ Proof.
Without loss of generality, we may assume that Π is the upper half plane. For a suitableconstant c , Green’s and Cauchy’s theorems imply that ż Π c c p w ´ y q m p x ´ w q m p w ´ y q m dm p w q “ ż B Π p w ´ y q m ` p x ´ w q m p w ´ y q m dw “ ż B Π p w ´ y q m ` p x ´ w q m p w ´ y q m dw “ . We will use an auxiliary function.
Definition 6.16.
Let us define h m p z q : “ ż B Ω p τ ´ z q m τ ´ z dτ for every z P Ω . By [Pra15b, Proposition 3.6] the weak derivatives of order m of h m are B j ¯ B m ´ j h m “ c m,j B j χ Ω , for 0 ď j ď m. (6.8)To shorten notation, we will write H jm “ B j h m .Combining (6.8) with Lemma 6.11, one obtains the following: Lemma 6.17.
Let ď j ď m , ă s ă , sp ą and let Ω be a bounded F s ` ´ p p,p -domain. Then ∇ m ´ j H jm P F sp,p p Ω q . Given a j times differentiable function f , we will write P jz p f qp ξ q “ ÿ | (cid:126)i |ď j D (cid:126)i f p z q (cid:126)i ! p ξ ´ z q (cid:126)i (6.9)for its j -th degree Taylor polynomial centered in the point z .29 roposition 6.18 (see [Pra15b, Proposition 3.6]) . Let Ω be a bounded Lipschitz domain, and let m ě . Then, for every pair z, ξ P Ω with z ‰ ξ , we have that K m p z, ξ q “ c m B B χ Ω p z q p ξ ´ z q m ´ p ξ ´ z q m ` ` ÿ j ď m c m,j H jm p ξ q ´ P m ´ jz H jm p ξ qp ξ ´ z q m ` ´ j , (6.10)Note that the Taylor polynomials are well defined because Lemma 6.17 implies the requireddifferentiability. Proof of Proposition 6.14.
We assume that p ě q , since otherwise, one has that } R m f } F sp,q p Ω q À} R m f } F sp,p p Ω q (see Proposition 5.4).Let 0 ă ρ ă f P F sp,q p Ω q , let us write D sq f p x q : “ ˜ż B p x,ρδ Ω p x qq | f p x q ´ f p y q| q | x ´ y | sq ` dy ¸ q . (6.11)We want to show that00 : “ ›› D sq R m f ›› L p p Ω q “ ˜ ÿ Q P W ›› D sq R m f ›› pL p p Q q ¸ p ď C h } f } C h p Ω q . For every Whitney cube Q we choose a bump function χ Q ď ϕ Q ď χ Q with | ∇ ϕ Q | ď (cid:96) p Q q .Then,00 p À ÿ Q | f Q | p ›› D sq R m ›› pL p p Q q ` ÿ Q ›› D sq R m p f ´ f Q q ϕ Q ›› pL p p Q q ` ÿ Q ›› D sq R m p f ´ f Q qp ´ ϕ Q q ›› pL p p Q q “ ` ` . (6.12)We will show that each term is bounded by C } f } p C h p Ω q .Let us begin by the first term in the right-hand side of (6.12), which is the easiest one. Indeed,for any cube Q , the mean | f Q | ď } f } L p Ω q . On the other hand the boundedness of R m in theTriebel-Lizorkin space under consideration implies that R m P F sp,q p Ω q . By Theorem 5.1, thisimplies 11 À } f } pL p Ω q } R m } pF sp,q p Ω q . Next, let us face the local part in (6.12). We fix the following notation: when dealing with thedifference of a function F between two points, we will write F rp x q ´ p y qs : “ F p x q ´ F p y q . Let x, y P Q P W . Then, since B r χ Ω c B m p χ Ω f qs is analytic on Ω, it has continuous derivatives and,thus, by the mean value theorem | R m rp f ´ f Q q ϕ Q srp x q ´ p y qs| ď }B R m rp f ´ f Q q ϕ Q s} L p Q q | x ´ y | and, fixing a convenient ρ in (6.11), we get that22 ď ÿ Q ż Q ˆż Q | R m rp f ´ f Q q ϕ Q srp x q ´ p y qs| q | x ´ y | sq ` dy ˙ pq dx ď ÿ Q }B R m rp f ´ f Q q ϕ Q s} pL p Q q ż Q ˆż Q | x ´ y | q | x ´ y | sq ` dy ˙ pq dx À ÿ Q (cid:96) p Q q `p ´ s q p }B R m rp f ´ f Q q ϕ Q s} pL p Q q . (6.13)30ake z P Q . Then B R m rp f ´ f Q q ϕ Q sp z q “ ż Ω c p z ´ w q ż Ω p w ´ ξ q m ´ p f p ξ q ´ f Q q ϕ Q p ξ qp w ´ ξ q m ` dm p ξ q dm p w q . It is immediate to check that this double integral is absolutely convergent and, thus, Fubini’sTheorem applies and it follows that B R m rp f ´ f Q q ϕ Q sp z q “ ż Q p f p ξ q ´ f Q q ϕ Q p ξ q ż Ω c p w ´ ξ q m ´ p z ´ w q p w ´ ξ q m ` dm p w q dm p ξ q . Next, we consider the half-plane Π Q from Lemma 6.7. Recall that dist p Q, Π cQ q « (cid:96) p Q q . Then,Lemma 6.15 implies that B R m rp f ´ f Q q ϕ Q sp z q “ ż Q p f p ξ q ´ f Q q ϕ Q p ξ q ˜ż Ω c ´ ż Π cQ ¸ p w ´ ξ q m ´ p z ´ w q p w ´ ξ q m ` dm p w q dm p ξ q . Since z P Q , taking absolute values we obtain |B R m rp f ´ f Q q ϕ Q sp z q| ď ż Q | f p ξ q ´ f Q | ż Ω∆Π Q | z ´ w | | w ´ ξ | dm p w q dm p ξ qÀ (cid:96) p Q q } f } L p Ω q ż Ω∆Π Q | w ´ z | dm p w q . By (6.5), we get |B R m rp f ´ f Q q ϕ Q sp z q| À (cid:96) p Q q } f } L p Ω q ¨˝ ÿ ρP Ą Q : (cid:96) p P qă R β p q p P q (cid:96) p P q ` R ´ . ˛‚ Back to (6.13), we have that22
À } f } pL p Ω q ÿ Q (cid:96) p Q q `p ´ s q p ¨˝¨˝ ÿ ρP Ą Q : (cid:96) p P qă R β p q p P q (cid:96) p P q ˛‚ p ` C ˛‚ . By Lemma 6.8 we get 22
À } f } pL p Ω q } ν } F s ´ pp,p pB Ω q . It remains to control the nonlocal part in (6.12), that is,33 ď ÿ Q ż Q ˆż Q | R m rp f ´ f Q qp ´ ϕ Q qsrp x q ´ p y qs| q | x ´ y | sq ` dy ˙ pq dx. As in (6.13), by the mean value property of analytic functions, we have that33 À ÿ Q (cid:96) p Q q `p ´ s q p }B R m rp f ´ f Q qp ´ ϕ Q qs} pL p Q q À ÿ Q (cid:96) p Q q p ´ p q`p ´ s q p }B R m rp f ´ f Q qp ´ ϕ Q qs} pL p Q q . (6.14)31ake z P Q . Then B R m rp f ´ f Q qp ´ ϕ Q qsp z q “ ż Ω c p z ´ w q ż Ω p w ´ ξ q m ´ p f p ξ q ´ f Q qp ´ ϕ Q p ξ qqp w ´ ξ q m ` dm p ξ q dm p w q . This double integral is absolutely convergent: ż Ω c ż Ω | f p ξ q ´ f Q || ´ ϕ Q p ξ q|| z ´ w | | w ´ ξ | dm p ξ q dm p w q À } f } L ż Ω c | log p δ Ω p w qq| ` | log p diam p Ω qq|| z ´ w | dm p w q . Thus, we can apply Fubini’s Theorem, (6.7) and (6.10) to get B R m rp f ´ f Q qp ´ ϕ Q qsp z q “ ż Ω z Q p f p ξ q ´ f Q qp ´ ϕ Q p ξ qq ż Ω c p w ´ ξ q m ´ p z ´ w q p w ´ ξ q m ` dm p w q dm p ξ q“ c m B B χ Ω p z q ż Ω z Q p f p ξ q ´ f Q qp ´ ϕ Q p ξ qq p ξ ´ z q m ´ p ξ ´ z q m ` dm p ξ q` ż Ω z Q p f p ξ q ´ f Q qp ´ ϕ Q p ξ qq ÿ j ď m c m,j p H jm p ξ q ´ P m ´ jz H jm p ξ qqp ξ ´ z q m ` ´ j dm p ξ q . Whenever z P Ω z supp F , we have that B m Ω F p z q “ c m ş Ω X supp F F p ξ q p ξ ´ z q m ´ p ξ ´ z q m ` dm p ξ q . Thus, we canapply this identity in the first term of the right-hand side above, and back to (6.14), we obtain33 À ÿ Q (cid:96) p Q q p ´ p q`p ´ s q p }B B χ Ω p z q B m Ω rp f ´ f Q qp ´ ϕ Q qsp z q} pL p Q,dm p z qq ` ÿ j ď m ÿ Q (cid:96) p Q q p ´ p q`p ´ s q p ›››››ż Ω z Q | f p ξ q ´ f Q | | H jm p ξ q ´ P m ´ jz H jm p ξ q|| ξ ´ z | m ` ´ j dm p ξ q ››››› pL p Q,dm p z qq “ ` m ÿ j “ ď ÿ Q (cid:96) p Q q `p ´ s q p }B B χ Ω } pL p Q q } B m Ω rp f ´ f Q qp ´ ϕ Q qs} pL p Q q . Using again the half-plane Π Q from Lemma 6.7, whose boundary minimizes β p q p Q q , for z P Q we can write B B χ Ω p z q “ c ˜ż Ω z B p z, δ Ω q ´ ż Π Q z B p z, δ Ω q ¸ p z ´ w q dm p w q (see [CT12, Lemma 4.2]), where we wrote again δ Ω p z q for dist p z, B Ω q . Taking absolute values, by(6.5) we get }B B χ Ω } L p Q q À ż Ω∆Π Q p w, Q q dm p w q À ÿ ρP Ą Q : (cid:96) p P qă R β p q p P q (cid:96) p P q ` R ´ . On the other hand, by [MOV09, Main Lemma] and doing some routine computations, one cancheck that } B m Ω rp f ´ f Q qp ´ ϕ Q qs} L p Ω q ď } B m Ω rp f ´ f Q qp ´ ϕ Q qs} C h p Ω q À m }p f ´ f Q qp ´ ϕ Q q} C h p Ω q À } f } C h p Ω q . ď } f } p C h p Ω q ÿ Q (cid:96) p Q q `p ´ s q p ¨˝ ÿ ρP Ą Q : (cid:96) p P qă R β p q p P q (cid:96) p P q ` R ´ ˛‚ p Using Lemma 6.8 again, 3.13.1 ď C } f } p C h p Ω q . Consider the term 3.2.j3.2.j in (6.15). We trivially control by the supremum norm of f :3.2.j3.2.j ď } f } pL p Ω q ÿ Q (cid:96) p Q q p ´ p q`p ´ s q p ›››››ż Ω z Q | H jm p ξ q ´ P m ´ jz H jm p ξ q|| ξ ´ z | m ` ´ j dm p ξ q ››››› pL p Q,dm p z qq . By Lemma 6.19 below, we get that3.2.j3.2.j
À } f } pL p Ω q ›› ∇ m ´ j H jm ›› F sp,p p Ω q , and by Lemma 6.17, the last factor is finite. The proof above depends on the following characterization:
Lemma 6.19.
Let n P N and ă s ă . Let Ω be a uniform domain with Whitney covering W ,and let F P C n p Ω q such that its weak derivatives ∇ n F P F sp,p p Ω q . ThenNN : “ ÿ Q P W (cid:96) p Q q p ´ p q`p ´ s q p ˜ż Q ÿ S P W ż S | F p ξ q ´ P nz F p ξ q| D p Q, S q n ` dm p ξ q dm p z q ¸ p À } ∇ n F } F sp,p p Ω q Meyers’ approximating polynomials are very useful to deal which such a situation: consider theset P n of polynomials of degree at most n . Given a cube Q and a function f P L p Q q , the Meyerspolynomial P nQ f P P n is the unique polynomial in P n satisfying that ş Q ∇ j f “ ş Q ∇ j P nQ f for j ď n . It satisfies the Poincar´e inequality ›› ∇ k p f ´ P nQ f q ›› L p Q q À (cid:96) p Q q n ´ k } ∇ n f ´ p ∇ n f q Q } L p Q q , (6.16)whenever f P W n, p Q q , k ď n . Proof of Lemma 6.19.
We change the Taylor polynomial centered at z P Q by the correspondingMeyers’ polynomial as follows:NN À ÿ Q P W (cid:96) p Q q ´ pp `p ´ s q p ¨˚˝ ÿ S P W ››››› P nQ F ´ P nz F ›› L p S q ››› L p Q,dm p z qq D p Q, S q n ` ˛‹‚ p ` ÿ Q P W (cid:96) p Q q ´ pp `p ´ s q p ˜ ÿ S P W (cid:96) p Q q ›› F ´ P nQ F ›› L p S q D p Q, S q n ` ¸ p “ EE ` MM . P ofdegree at most n and disjoint cubes Q and S , we have that } P } L p S q À (cid:96) p S q D p Q, S q n (cid:96) p Q q ` n } P } L p Q q . (6.17)(use the fact that all norms on P n are equivalent and appropriate rescaling factors). Thus,EE À ÿ Q P W (cid:96) p Q q ´ pp `p ´ s q p ¨˚˝ ÿ S P W ››››› P nQ F ´ P nz F ›› L p Q q ››› L p Q,dm p z qq D p Q, S q (cid:96) p S q (cid:96) p Q q ` n ˛‹‚ p Using Fubini, we can change the order of integration and since the Taylor polynomial of apolynomial of the same degree is itself, we get ››››› P nQ F ´ P nz F ›› L p Q q ››› L p Q,dm p z qq ď ››››› P nz p F ´ P nQ F q ›› L p Q,dm p z qq ››› L p Q q (cid:96) p Q q . But using the expression (6.9) of the Taylor Polynomial of degree n , for ξ, z P Q we have that } P nz f p ξ q} L p Q,dm p z qq ď ÿ ď| (cid:126)i |ď n (cid:126)i ! ››› D (cid:126)i f p z qp ξ ´ z q (cid:126)i ››› L p Q,dm p z qq À n ÿ k “ ›› ∇ k f ›› L p Q q (cid:96) p Q q k . Plugging the Poincar´e inequality (6.16) in, we get ››››› P nQ F ´ P nz F ›› L p Q q ››› L p Q,dm p z qq À (cid:96) p Q q ` n } ∇ n F ´ p ∇ n F q Q } L p Q q . Back to the error term, we get thatEE À ÿ Q P W (cid:96) p Q q ´ pp `p ´ s q p ˜ } ∇ n F ´ p ∇ n F q Q } L p Q q ÿ S P W (cid:96) p S q D p Q, S q ¸ p . Using (2.1) and the H¨older inequality,EE À ÿ Q P W (cid:96) p Q q ´ pp `p ´ s q p } ∇ n F ´ p ∇ n F q Q } pL p p Q q (cid:96) p Q q pp (cid:96) p Q q ´ p , so EE À ÿ Q P W } ∇ n F ´ p ∇ n F q Q } pL p p Q q (cid:96) p Q q sp À } ∇ n F } F sp,p p Ω q . To estimate the main term MM we will argue by duality. Writing h Q : “ (cid:96) p Q q ´ s ` p ÿ S P W ›› F ´ P nQ F ›› L p S q D p Q, S q n ` , it follows thatMM p “ }t h Q u Q P W } (cid:96) p p W q “ ˜ sup t g Q u ÿ Q P W h Q g Q ¸ “ sup t g Q u ÿ Q P W (cid:96) p Q q ´ s ` p g Q ÿ S P W ›› F ´ P nQ F ›› L p S q D p Q, S q n ` t g Q u Q P W satisfying that }t g Q u Q P W } (cid:96) p p W q “ . Fix Whitney cubes Q and S . Next we use a telescoping summation following the chain of cubes r Q, S s introduced in Definition 2.4: ›› F ´ P nQ F ›› L p S q ď } F ´ P nS F } L p S q ` ÿ P Pr Q,S q ››› P nP F ´ P n N p P q F ››› L p S q , where N p P q stands for the next cube in the chain r Q, S s . By Definition 2.2, the side of a givenWhitney cube is at most twice as long as the side of its neighbors. Thus, for P P r
Q, S s we have P Ă N p P q . Using (6.17), ›› F ´ P nQ F ›› L p S q À ÿ P Pr Q,S s } F ´ P nP F } L p P q (cid:96) p S q D p P, S q n (cid:96) p P q ` n . Using the Poincar´e inequality (6.16) and the H¨older inequality, for P P r
Q, S s we get that } F ´ P nP F } L p P q À } ∇ n F ´ p ∇ n F q P } L p P q (cid:96) p P q n À } ∇ n F ´ p ∇ n F q P } L p p P q (cid:96) p P q n ` p À ›› D sp ∇ n F ›› L p p P q (cid:96) p P q s ` n ` p (see (6.11)). Since D p P, S q À D p Q, S q , we obtainMM p À sup t g Q u ÿ P P W ›› D sp ∇ n F ›› L p p P q ÿ Q,S : P Pr Q,S s (cid:96) p Q q ` p ´ s g Q (cid:96) p S q (cid:96) p P q p ´ s D p Q, S q . To end the proof we need will use the boundedness of the maximal Hardy-Littlewood operatoron Lebesgue spaces, so we define G p x q : “ ř Q P W g Q (cid:96) p Q q { p χ Q p x q . It is clear that } G } L p p Ω q “}t g Q u Q P W } (cid:96) p p W q “
1, and (cid:96) p Q q p g Q “ ş Q G . Thus,MM p À sup G P L p p Ω q ÿ P P W ›› D sp ∇ n F ›› L p p P q ÿ Q,S : P Pr Q,S s (cid:96) p Q q ´ s (cid:96) p S q (cid:96) p P q p ´ s D p Q, S q ż Q G p x q dx. By Lemma 6.20 below (take (cid:96) “ p À sup G P L p p Ω q ÿ P P W ›› D sp ∇ n F ›› L p p P q ›› M G ` M G ›› L p p P q À ›› D sp ∇ n F ›› L p p Ω q . Here M stands for the maximal Hardy-Littlewood operator, which is bounded in L p .It remains to show the following version of [Pra15b, Lemma 2.5]. Lemma 6.20.
Consider a uniform domain Ω Ă R d with Whitney covering W , a cube P P W , afunction G P L p Ω q and two real numbers ă s ă (cid:96) . Then ÿ Q,S : P Pr Q,S s (cid:96) p Q q (cid:96) ´ s (cid:96) p S q D p Q, S q ` (cid:96) ż Q G p x q dx À (cid:96) p P q s ż P p M G p x q ` M G p x qq dx.. roof. We divide the chain r Q, S s “ r
Q, Q S s Y r Q S , S s , in such a way that if P is in the ascendingpath r Q, Q S s then (cid:96) p P q « D p Q, P q and D p P, S q « D p Q, S q « (cid:96) p Q S q , and if P is in the descendingpath we get analogous conditions, see (2.3). Thus, we write ¨˝ ÿ Q,S : P Pr Q,Q S s ` ÿ Q,S : P Pr Q S ,S s ˛‚ (cid:96) p Q q (cid:96) ´ s (cid:96) p S q D p Q, S q ` (cid:96) ż Q G p x q dx “ : AA ` DD . For every cube P we get ÿ Q :D p Q,P qÀ (cid:96) p P q (cid:96) p Q q (cid:96) ´ s ż Q G p x q dx ď (cid:96) p P q (cid:96) ´ s ż P M G p x q dx. (6.18)In the ascending path, thus, using (2.1) and (6.18) we obtainAA ď ÿ S P W (cid:96) p S q D p S, P q ` (cid:96) ÿ Q :D p Q,P qÀ (cid:96) p P q (cid:96) p Q q (cid:96) ´ s ż Q G p x q dx « (cid:96) p P q ´ (cid:96) (cid:96) p P q (cid:96) ´ s ż P M G p x q dx In the descending path, we divide the sum in Q in “dyadic annuli” just by setting R “ Q S :DD “ ÿ S :D p S,P qÀ (cid:96) p P q (cid:96) p S q ÿ R :D p R,P qÀ (cid:96) p R q (cid:96) p R q ` (cid:96) ÿ Q :D p Q,R qÀ (cid:96) p R q (cid:96) p Q q (cid:96) ´ s ż Q G p x q dx. Again, first we will use (6.18) to getDD À (cid:96) p P q ÿ R :D p R,P qÀ (cid:96) p R q (cid:96) p R q ` s ż R M G p x q dx À (cid:96) p P q ÿ R :D p R,P qÀ (cid:96) p R q (cid:96) p R q s (cid:96) p P q ż P M G p y q dy. Since the last sum above is a geometric series, we obtain thatDD À (cid:96) p P q s ż P M G p y q dy. A Appendix
A.1 Composition and inverse function theorems
First we need a lemma on a generalized chain rule. For this purpose we recover the multivariateversion of Fa`a di Bruno’s formula (see [KP92, Lemma 1.3.1] for the one-dimensional case), whoseproof is a mere exercise on induction. Given a multiindex (cid:126)i P N d , we define m p (cid:126)i q P t , ¨ ¨ ¨ , d u | (cid:126)i | asthe vector whose components are non-decreasing (i.e, m p (cid:126)i q (cid:96) ď m p (cid:126)i q (cid:96) ` ), and such that t j : m p (cid:126)i q (cid:96) “ j u “ (cid:126)i j . For instance, m p , q “ p , , , , q , and m p , , q “ p , , , , q .36 emma A.1 (Chain rule) . Given f “ p f , ¨ ¨ ¨ , f d q : R d Ñ R d , f : R d Ñ R with f i , f P W M, p R d q and (cid:126)k P N d with | (cid:126)k | “ M , there exist appropriate constants such that D (cid:126)k p f ˝ f q “ ÿ ď| (cid:126)i |ď M t α j u | (cid:126)i | j “ Ă N d zt (cid:126) u : ř | α j |“ M C (cid:126)k,(cid:126)i, t α j u D (cid:126)i f p f q | (cid:126)i | ź (cid:96) “ D α (cid:96) f m p (cid:126)i q (cid:96) (A.1) almost everywhere. Most likely the following results appear in the literature, but we were not able to locate them,so we include these results for the sake of completeness. First we apply the chain rule to H¨olderfunctions to show Lemma 4.1.
Lemma A.2.
Let ă s , s R N and d P N . Given bounded Lipschitz domains Ω j Ă R d andfunctions f j P C s p Ω j q with f p Ω q Ă Ω and f bi-Lipschitz, then f ˝ f P C s p Ω q , (A.2) and f ´ P C s p Ω q . (A.3) Proof.
Let us check (A.2). According to (A.1), for s “ k ` σ with k P N , 0 ă σ ă
1, we get | ∇ k p f ˝ f qp x q ´ ∇ k p f ˝ f qp y q|À ÿ ď i ď k | ∇ i f p f p x qq ´ ∇ i f p f p y qq| ÿ t α j u : ř i α j “ k i ź j “ | ∇ α j f p x q| (A.4) ` ÿ ď i ď k | ∇ i f p f p y qq| ÿ t α j u : ř i α j “ k i ÿ (cid:96) “ | ∇ α (cid:96) f p x q ´ ∇ α (cid:96) f p y q| (cid:96) ´ ź j “ | ∇ α j f p y q| i ź j “ (cid:96) ` | ∇ α j f p x q| . This implies that } f ˝ f } C s À ÿ ď i ď k C f ›› ∇ i f ›› C σ ÿ t α j u : ř i α j “ k i ź j “ } ∇ α j f } L ` ÿ ď i ď k ›› ∇ i f ›› L ÿ t α j u : ř i α j “ k i ÿ (cid:96) “ } ∇ α (cid:96) f } C σ ź j ‰ (cid:96) } ∇ α j f } L , so } f ˝ f } C s ď C f } f } C s p Ω q , (A.5)with C f depending polynomially on the C σ norm of the derivatives of f and its bi-Lipschitzconstant.Finally, let us prove (A.3). Writing f “ f and applying the inverse function theorem, D p f ´ qp x q “ p Df q ´ p f ´ p x qq . That is, the first-order derivatives of the inverse can be expressed as p D p f ´ qq ij “ g ij ˝ p f ´ q , where g ij “ P ij p DF q det p Df q (A.6)37or certain homogeneous polynomials P ij : R d ˆ d Ñ R of degree d ´
1. By induction we can assume f ´ P C s ´ , and by (A.2) it is enough to check that g ij P C s ´ p Ω q . But every derivative of degree k ´ g ij is a polynomial on the derivatives of f with at most one factor of order k at each term,divided by a power of the Jacobian determinant. Applying the argument in (A.4) to each of thesederivatives we obtain (A.3).Next we adapt the approach to show Lemma 3.1 on Sobolev spaces. The results are valid forspaces W s,p whose functions have all the derivatives bounded except perhaps the last ones. Thus,we define the space W s,p p Ω q : “ W s,p p Ω q X W s ´ , p Ω q . Note that W , stands for L . By theSobolev embedding Theorem, when p ą d we have that W s,p p Ω q “ W s,p p Ω q . Lemma A.3.
Let s, d P N , and ă p ď 8 . Given bounded Lipschitz domains Ω j Ă R d andfunctions f j P W s,p p Ω j q with f p Ω q Ă Ω and f bi-Lipzchitz, then f ˝ f P W s,p p Ω q , (A.7) and f ´ P W s,p p Ω q . (A.8) Proof.
To check (A.7), the case s “ s ě
2. Sinceboth f and f are in W s ´ , p Ω j q (that is, they have Lipschitz derivatives up to order s ´ f j are W s, p Ω j q functions. By (A.1), and applying H¨older’s inequality,we get that } ∇ s p f ˝ f q} L p À d,s ÿ ď i ď s ÿ t α k u i : ř α k “ s ›› ∇ i f p f q ›› p i ź (cid:96) “ } ∇ α (cid:96) f } p (cid:96) , where ř i p (cid:96) “ p .Consider the term with i “ s , so α (cid:96) “ ď (cid:96) ď s . In this case, take p “ p , p (cid:96) “ 8 . Next,consider the other end-point i “
1, where there is only one ∇ α (cid:96) f factor and α “ s . In this caseone can take p “ 8 and p “ p because ∇ f P L p Ω j q . The intermediate terms (1 ă i ă s ) arebounded analogously, with more freedom in the choice of p (cid:96) . Thus, } ∇ s p f ˝ f q} L p À C f } f } W s,p p Ω q , (A.9)with the constant C f depending polynomially on the W ,p norm of the derivatives of f up toorder s ´ f j P W s,p p Ω j q with f bi-Lipschitz. Let Λ Ω j be an extensionoperator for W s,p p Ω j q (see [Ste70, Theorem VI.5]). Consider approximations of the identity f εj “ Λ Ω j f j ˚ ϕ ε for a C c function ϕ with ş ϕ “ f εj Ñ f j in W s,p p Ω j q . In particular, the first order derivatives of f εj converge uniformlyto the corresponding derivatives of f j and, thus, the bi-Lipschitz character is preserved for ε smallenough. Thus, estimate (A.9) holds uniformly, and via Banach-Alaoglu Theorem, it extends to thelimiting case.The inverse function bound (A.8) can be proven by the same methods using (A.6).Finally we see that Triebel-Lizorkin spaces have the same properties, proving Lemma 5.6,Lemma 3.2 being a particular case. Again, we define F sp,q p Ω q : “ F sp,q p Ω q X W k, p Ω q for k ă s ă k `
1. By Proposition 5.4 and (2.12), when p s ´ k q p ą d we have that F sp,q p Ω q “ F sp,q p Ω q .38 emma A.4. Let k P N , ă σ ă and let s : “ k ` σ , let ă p, q ă 8 and d P N with σ ą dp ´ dq .Given bounded Lipschitz domains Ω j Ă R d and functions f j P F sp,q p Ω j q with f p Ω q Ă Ω and f bi-Lipzchitz, then f ˝ f P F sp,q p Ω q (A.10) and f ´ P F sp,q p Ω q . (A.11)Note that we do not include the end-point cases of Lemma 3.2: the case p “ q “ 8 coincideswith the H¨older spaces studied in Lemma A.2, the case p “ q “ Proof.
We begin by showing (A.10). Since the case k “ k ě
1. Also by Theorems 5.1 and 5.2, it is enough to check that1 : “ ż Ω ˆż Ω | ∇ k p f ˝ f q p x q ´ ∇ k p f ˝ f q p y q| q | x ´ y | σq ` d dy ˙ pq dx ď C pf } f } p F sp,q p Ω q . (A.12)We can use the chain rule (A.1) almost everywhere and in particular (A.4) applies. We obtain1 À ÿ ď i ď k t α j u : ř i α j “ k i ź j “ } ∇ α j f } pL ż Ω ˆż Ω | ∇ i f p f p x qq ´ ∇ i f p f p y qq| q | x ´ y | σq ` d dy ˙ pq dx (A.13) ` ÿ ď i ď k t α j u : ř i α j “ k ď (cid:96) ď i ›› ∇ i f ›› pL ź j ‰ (cid:96) } ∇ α j f } pL ż Ω ˆż Ω | ∇ α (cid:96) f p x q ´ ∇ α (cid:96) f p y q| q | x ´ y | σq ` d dy ˙ pq dx. Using f as a bi-Lipschitz change of variables for the first integral in the right-hand side of thelast inequality above, we show (A.10). In particular, one obtains (A.12), with the constant C f depending polynomially on the L and F σp,q norms of the derivatives of f and on its bi-Lipschitzconstant.Inequality (A.11) is proven by analogous techniques. A.2 Trace spaces
The last part of the appendix is devoted to making sure that all the definitions in the literaturethat we use coincide. We begin by a toy question: the definition of the trace spaces does notdepend on the particular parameterization we choose.Whenever Ω is a planar Lipschitz domain there exist M boundary points x j P B
Ω, a radius R ą δ -Lipschitz functions A j : R Ñ R such that the following holds: • Locally, the boundary is described byΩ X B p x j , R q “ t x P B p x j , R q : π j p x q ą A j p π j p x qqu , (A.14)where π j p x q “ x x ´ x j , e j y is the projection to an appropriate line passing through x j and π j p x q “ x x ´ x j , e j y is the projection to the perpendicular line, i.e., e j is the unit normalvector orthogonal to e j pointing into the domain. If the boundary parameterizations aresmooth we can assume that e j is a unit tangent vector at x j and e j the unit inward normalvector. 39 The boundary is covered by B Ω Ă ď B ˆ x j , ε δ ˙ (A.15)for ε δ : “ R ? ` δ , which grants that whenever | x | ă ε δ , the image p x, A j p x qq under a parame-terization A j is a boundary point even if the Lipschitz constant is big. • The balls B ` x j , ε δ ˘ are pairwise disjoint.From Remark 2.10), we know that a F sp,p -domain with p s ´ q p ą p δ, R q -Lipschitz domain(with R , δ and M above depending on the minimum Besov norm of γ j and the distance betweenboundary components), and its parameterizations are in C s ´ p Ă C k for s “ k ` σ , k P N and0 ă σ ă σp ą
1. In the following lemmata, we will deal with a larger class of domains,dropping the restriction σp ą
1. Then we need to assume continuity of the derivatives from thebeginning. Again we write F sp,p : “ F sp,p X W k, , and we say that Ω is a F sp,p -domain if it is aLipschitz domain and it has parameterizations in F sp,p . Note that, in particular, Ω is bounded. Lemma A.5.
Let s ą be non-integer, let ď p ď 8 , and let Ω be a F sp,p -domain. Let γ , γ be two bi-Lipschitz, F sp,p parameterizations of the same component of the boundary and let f : B Ω Ñ C . Then } f ˝ γ } F sp,p p T q « } f ˝ γ } F sp,p p T q . Proof.
By elementary manipulations and (A.12) (or (A.5) when p “ 8 ), we have that } f ˝ γ } F sp,p p T q “ ›› p f ˝ γ q ˝ ` γ ´ ˝ γ ˘›› F sp,p p T q À C γ ´ ˝ γ } f ˝ γ } F sp,p p T q , as long as we can show that ›› γ ´ ˝ γ ›› F sp,p p T q ă C .Let t ψ j u Mj “ be a partition of the unity given by C bump functions so that ψ j p x q “ γ p x q R B p x j , ε δ q . Then, we can write γ ´ ˝ γ “ M ÿ j “ ψ j γ ´ ˝ γ “ M ÿ j “ ψ j p π j ˝ γ q ´ ˝ p π j ˝ γ q . Note the the last term above must be understood in the sense that p π j ˝ γ q ´ ˝p π j ˝ γ q is computedonly when ψ j ‰
0. Using (3.6) and Lemma A.4 (or (4.1) and Lemma A.2 in the p “ 8 case) weget γ ´ ˝ γ P F sp,p p T q . The control on the W k, norm comes from (3.1) and Lemma A.3. Lemma A.6.
Let s ą be non-integer, let ď p ď 8 , and let Ω be a F sp,p -domain. Then, the arcparameterization of the boundary is in F sp,p p T q .Proof. Let γ be a bi-Lipschitz F sp,p p T q parameterization of a component of the boundary. Then,we define the reparameterization τ p t q : “ ż t | γ p x q| dx. Note that τ p π q is the length of the curve (cid:96) p γ q . It is well-known that γ p t q : “ γ ˝ τ ´ ´ (cid:96) p γ q π t ¯ is the arc parameterization of the boundary component, i.e., | γ p t q| “ (cid:96) p γ q π . To show that γ isbi-Lipschitz and γ P F sp,p p T q we will check that τ is bi-Lipschitz and τ P F sp,p p T q . (A.16)Once this is shown, the lemma follows using Lemma A.4 (Lemma A.2 in the p “ 8 case).40s a matter of fact, τ is bi-Lipschitz because γ is: whenever t ą t we get | τ p t q ´ τ p t q| “ ż t t | γ p x q| dx « t ´ t . Thus, by elementary embeddings τ P F σp,p p T q where s “ k ` σ for k P N and 0 ă σ ă
1. By (3.9),we only need to check that τ P F s ´ p,p p T q .If k ě τ is bounded by assumption and since τ p t q “ a γ p t q ` γ p t q , using expression(A.12) and the bi-Lipschitz character of τ it is a routine computation to check that its Besov normis bounded, so τ P F σp,p p T q , settling the case k “ k ě τ p k q p t q “ c k γ p k q γ ` γ p k q γ τ ` g k , where g k is a polynomial combination of derivatives of γ and τ of lower order. Therefore g k P F σp,p p T q by induction and the algebra structure of these spaces, and the lemma follows by thecharacterization of the Besov space in terms of differences.The previous lemma has as a consequence that the domains appearing in [Pra17] are exactlythe ones in Definition 3.3: Proposition A.7.
Let s ą be non-integer and let ď p ď 8 . A bounded Lipschitz domain Ω isa F s ` p,p -domain if and only if given its outward unit normal vector ν and every arc parameterization γ of a component of its boundary, we have that ν ˝ γ P F sp,p p T q .Proof. Let Ω be a F s ` p,p -domain. Then, by Lemma A.6 we have that the arc parameterization γ is bi-Lipschitz and γ P F s ` p,p p T q . Thus, ν ˝ γ p t q “ ˘ iγ p t q is in F sp,p p T q .For the converse, we just need to see that the arc parameterization γ of a boundary componentΓ Ă B
Ω is bi-Lipschitz and it belongs to F s ` p,p p T q . The first comes from the fact that the domainis Lipschitz. For the Besov character, } γ } F s ` p,p p T q « } γ } L p p T q ` ›› γ ›› F sp,p p T q “ H p Γ q p ` } iν ˝ γ } F sp,p p T q . (A.17)The W k ` , p T q character is controlled mutatis mutandis.Combining the approaches of both lemmas above, we get the following: Lemma A.8.
Let s ą be non-integer, let ď p ď 8 , let Ω be a F sp,p -domain with M , δ , R , x j , A j and π (cid:96)j defined so that (A.14) and (A.15) are satisfied and let γ P F sp,p p T q be a bi-Lipschitzparameterization of a boundary component. Then, for every f P L pB Ω q , we have that } f ˝ γ } F sp,p p T q « M ÿ j “ ›› f ˝ p π j q ´ ›› F sp,p p ´ ε δ , ε δ q , with constants independent of f .In particular, the norms under different choices of the boundary points x j are equivalent.Proof. Let t ψ j u Mj “ be a partition of the unity given by C bump functions so that ψ j p t q “ γ p t q R B p x j , ε δ q . Note that for every j ď M , we have that ψ j p t q f ˝ γ p t q “ ψ j p t q ` f ˝ p π j q ´ ˘ ˝ ` π j ˝ γ ˘ p t q ´ ε δ ď t ď ε δ . Thus, the first inequality } f ˝ γ } F sp,p p T q À M ÿ j “ ›› ψ j ` f ˝ p π j q ´ ˘ ˝ ` π j ˝ γ ˘›› F sp,p p T q À M ÿ j “ ›› f ˝ p π j q ´ ›› F sp,p p ´ ε δ , ε δ q follows as in the proof of Lemma A.5, with constants depending on γ and the choice of boundarypoints.On the other hand, let t r ψ j u Mj “ be a partition of the unity given by C bump functions so that r ψ j p t q “ p π j q ´ p t q R B p x j , ε δ q and r ψ j p t q “ p π j q ´ p t q P B p x j , ε δ q . Then,for every j ď M we can write r ψ j f ˝ p π j q ´ “ r ψ j p f ˝ γ q ˝ ` π j ˝ γ ˘ ´ . Thus, M ÿ j “ ›› f ˝ p π j q ´ ›› F sp,p p ´ ε δ , ε δ q ď M ÿ j “ ››› r ψ j p f ˝ γ q ˝ ` π j ˝ γ ˘ ´ ››› F sp,p p ´ ε δ , ε δ q and we get M ÿ j “ ›› f ˝ p π j q ´ ›› F sp,p p ´ ε δ , ε δ q À } f ˝ γ } F sp,p p T q , where we have used (3.6), (3.11) and (A.12) again and the finite overlapping of the balls. Theconstants depend on γ and the choice of boundary points. Finally the W k, p T q character by thesame scheme but using Lemma A.3.To end, we get the following: Proposition A.9.
Let k P N , k ă s ă k ` , ď p ď 8 . Let Ω be a W k ` , -domain and let M , δ , R , x j , A j and π (cid:96)j be defined so that (A.14) and (A.15) are satisfied. Then, ˜ ÿ Q P W β p k q p Q q p (cid:96) p Q q ´ sp ¸ p ` H pB Ω q p « } ν } F sp,p pB Ω q , with β p k q p Q q as in Definition 6.4. If p s ´ p q k ą the constants depend only on δ , R and theconstants of the Whitney covering. Otherwise they also depend on the W k ` , -character of thedomain. Here we understand F sp,p pB Ω q as functions defined on B Ω that are in F sp,p p T q when composedwith the arc parameterization of each boundary component. This proposition was already shownin [Pra17, Lemma A.1], but it comes easily from Lemmas 6.3 and A.8 (choosing f “ Id ). Proof.
By means of a translation we may assume that 0 P Ω.Let γ be the arc parameterization of a boundary component. If the sum of beta coefficients ´ř Q P W β p k q p Q q p (cid:96) p Q q ´ sp ¯ p ă 8 , then the boundary parameterizations A j P F s ` p,p by Lemma 6.3.Writing τ j p t q : “ ż t b ` A j p x q dx, then τ j is bi-Lipschitz, and estimate τ j P F s ` p,p p Id p¨q e j ` A j p¨q e j q ˝ τ ´ j isparameterized by the arc. Thus, in an appropriate interval J j Ă T there is an affine transformation φ j such that γ p t q “ p Id p¨q e j ` A j p¨q e j q ˝ τ ´ j ˝ φ j p t q for t P J j .Abusing notation we assume that φ j is the identity, leaving the necessary modifications to thereader. We can write ν ˝ γ “ ˘ iγ “ c j p A j p¨q e j ´ e j q ˝ τ ´ j p τ ´ j q , so ν ˝ γ is in F sp,p p J j q by Lemma A.4 and the algebra structure of the space (use Lemma A.2 if p “ 8 ). Since the intervals J j cover T , and } ν ˝ γ } F sp,p p T q is finite.Thus, we may assume that } ν ˝ γ } F sp,p p T q is finite. Using (A.17) and Lemma A.8 (choosing f “ Id ), we obtain } ν ˝ γ } F sp,p p T q ` diam p Ω q p « } γ } F s ` p,p p T q « m ÿ j “ ›› x j ` Id p¨q e j ` A j p¨q e j ›› F s ` p,p p ´ ε δ , ε δ q« m ÿ j “ } A j } F s ` p,p p ´ ε δ , ε δ q ` H pB Ω q p , where we assume that the boundary component under consideration is covered by the balls indexedfor 1 ď j ď m . By Lemma 6.3 and using some appropriate bump functions again, we get } ν } F sp,p pB Ω q ` diam p Ω q p « ˜ ÿ Q P W β p k q p Q q p (cid:96) p Q q ´ sp ¸ p ` } A j } W k ` , p ´ ε δ , ε δ q ` H pB Ω q p . Noting that } ν } L p “ H pB Ω q p ě diam p Ω q p , we get } ν } F sp,p pB Ω q ` } ν } W k, pB Ω q « ˜ ÿ Q P W β p k q p Q q p (cid:96) p Q q ´ sp ¸ p ` ÿ j } A j } W k ` , p ´ ε δ , ε δ q ` H pB Ω q p , with constants depending only on δ , R and the constants of the Whitney covering.The case p “ 8 holds as well in the preceding lemma and in the proposition, as long as LemmaA.2 holds true in the range of indices. In particular, for non-integer s P R ` ` sup Q P W β p k q p Q q (cid:96) p Q q s « } ν } C s pB Ω q . Acknowledgements
The first author was supported by Academy of Finland project SA13316965. The secondauthor was supported by the ERC grants 320501 (FP7/2007-2013) and 307179-GFTIPFD,and partially supported by MTM-2016-77635-P (Spain) and 2017-SGR-395 (Catalonia). Thethird author was supported by the Academy of Finland CoE “Analysis and Dynamics”, as wellas the Academy of Finland Project “Conformal methods in analysis and random geometry”.
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