Boundary Value Problems for the Stokes System in Arbitrary Lipschitz Domains

Transcription

1 Boundary Value Problems for the Stokes System in Arbitrary Lischitz Domains Marius Mitrea and Matthew Wright Contents 1 Introduction Descrition of main well-osedness results Consequences of the solvability of the inhomogeneous roblem Smoothness saces and Lischitz domains Grah Lischitz domains Hardy saces on grah Lischitz surfaces Bounded Lischitz domains Besov and Triebel-Lizorkin saces in Lischitz domains Smoothness saces on Lischitz boundaries Rellich identities for divergence form, second-order systems Green formulas A general Rellich identity for second order systems The Stokes system and hydrostatic otentials Bilinear forms and conormal derivatives Hydrostatic layer otential oerators Traces of hydrostatic layer otentials in Hardy saces Integral reresentation formulas Boundary integral oerators and the transmission roblem The L transmission roblem with near Rellich identities and related estimates The case of a grah Lischitz domain Inverting the double layer on L for near 2 on bounded Lischitz domains Inverting the single layer on L for near 2 on bounded Lischitz domains L -boundary value roblems on bounded Lischitz domains for near Local L 2 estimates Pressure, Cacciooli, and local boundary estimates Reverse Hölder estimates The transmission roblem in two and three dimensions Uniqueness Math Subject Classification. Primary: 35J25, 42B20, 46E35. Secondary 35J05, 45B05, 31B10. Key words: Stokes system, Lischitz domains, boundary roblems, layer otentials, Besov-Triebel-Lizorkin saces The research of the authors was suorted in art by the NSF 1

2 7.2 Atomic estimates Interolation arguments Higher dimensions Preliminary estimates The Dirichlet roblem Boundary value roblems in bounded Lischitz domains Localization arguments Main well-osedness results with nontangential maximal function estimates The Poisson roblem for the Stokes system Stokes-Besov and Stokes-Triebel-Lizorkin saces Conormal derivatives on Stokes-Besov and Stokes-Triebel-Lizorkin scales The conormal derivative of the Stokes-Newtonian otentials The conormal on Besov and Triebel-Lizorkin saces: the general case Layer otentials on Besov and Triebel-Lizorkin saces The Poisson roblem with Dirichlet and Neumann boundary conditions Aendix Smoothness saces in the Euclidean setting Gehring s lemma Hole-filling lemma Korn s inequality Hardy s estimate Traces in Hardy saces Saces of null-solutions of ellitic oerators Singular integral oerators on Sobolev-Besov saces Functional analysis on quasi-banach saces Surface to surface change of variables Truncating singular integrals Aroximating Lischitz domains References 221 Abstract The goal of this work is to treat the main boundary value roblems for the Stokes system, i.e., i) the Dirichlet roblem with L -data and nontangential maximal function estimates, ii) the Neumann roblem with L -data and nontangential maximal function estimates, iii) the Regularity roblem with L 1-data and nontangential maximal function estimates, iv) the transmission roblem with L -data and nontangential maximal function estimates, v) the Poisson roblem with Dirichlet condition in Besov-Triebel-Lizorkin saces, vi) the Poisson roblem with Neumann condition in Besov-Triebel-Lizorkin saces, in Lischitz domains of arbitrary toology in R n, for each n 2. Our aroach relies on boundary integral methods and yields constructive solutions to the aforementioned roblems. 2

3 1 Introduction 1.1 Descrition of main well-osedness results Informally seaking, the goal of the resent work is to rove otimal well-osedness results for homogeneous and inhomogeneous) boundary-value roblems for the Stokes system in Lischitz domains with arbitrary toology, in all sace dimensions and for all major tyes of boundary conditions Dirichlet, Neumann, transmission). The boundary data is selected from Lebesgue, Sobolev, Hardy, Besov and Triebel-Lizorkin saces and the smoothness of the solutions is measured accordingly. At the core of our analysis is the transmission roblem for the Stokes system, on which we wish to elaborate first. Let Ω be a Lischitz domain in R n, n 2, and define Ω + := Ω and Ω = R n \ Ω. The transmission boundary value roblem for the Stokes system studied here is of the tye u ± = π ± in Ω ±, T µ ) div u ± = 0 in Ω ±, u + u = g L 1 ), λ ν u +, π + ) µ λ ν u, π ) = f L ), M u ± ), Mπ ± ) L ). Here, is the Lalacian, µ [0, 1) is a fixed arameter, and ν := ν + is the outward unit normal to Ω +. For 1 < <, L 1 ) is the classical L -based Sobolev saces of order one on, M denotes the non-tangential maximal oerator cf. 2.5)), and 1.1) λ ν u ±, π ± ) := u ± + λ u ± ) ν π ± ν 1.2) is a family of co-normal derivatives, indexed by a arameter λ R more detailed definitions are given in subsequent chaters). In this way, we can simultaneously treat various tyes of Neumann boundary conditions. For examle, when λ = 0, 1.2) corresonds to the co-normal derivative treated in [34], whereas when λ = 1, 1.2) corresonds to the sli condition considered in [23]. Two closely related boundary value roblems are the Neumann roblem and the Dirichlet roblem with maximally) regular data: u = π in Ω, u = π in Ω, div u = 0 in Ω, div u = 0 in Ω, N) ν λ u, π) = f R) L ), u = g L 1 ), 1.3) M u), Mπ) L ) M u), Mπ) L ). From this oint forth, we will refer to R) as the Regularity roblem. Fabes, Kenig, and Verchota roved in [34] that N) and R) are well-osed if 2 ε < < 2 + ε, where ε = ε) > 0. Building on the work in [21], [75], Z. Shen has established in [82] a weak maximum rincile for the Dirichlet roblem for the Stokes system in Lischitz domains in R 3. Interolating this L bound with the L -estimates from [34] with near 2 shows that the Dirichlet roblem for the Stokes system in three-dimensional Lischitz domains with data in L is solvable whenever 2 ε < <. However, as ointed out by P. Deuring on. 16 of [29], this leaves oen the question of whether these solutions 1

4 may be constructed by means of the boundary layer method, and how to deal with exterior roblems and sli boundary conditions. With these aims in mind, let us briefly discuss the relevance of the transmission roblem itself. From a hysical oint of view, the transmission roblem µ ± u ± = π ± in Ω ±, T ) div u ± = 0 in Ω ±, u + u = g, σ λ u + σ λ u = f, 1.4) where σ λ u ± := µ ± u ± + λ u ± ) ν π ± ν, 1.5) describes the flow of a viscous incomressible fluid within and around a stationary article occuying the domain Ω + which is further embedded into a second orous medium Ω. In this context, u + and π + are the volume-averaged fluid velocity and ressure fields of the inner flow, whereas u and π have analogous roles for the outer flow. In the secific case when λ = 1, this is a standard roblem that arises when studying the low Reynolds number deformation of a viscous dro immersed in another fluid see [78]; [76], Sec. 7.2). Here, µ + denotes the viscosity of the dro, while µ denotes the viscosity of the surrounding fluid. The case when g = 0 is often of articular interest, since this introduces the hysically relevant restriction that the velocities u + and u must match on the boundary. The reader is referred to M. Kohr and I. Po s monograh [55] for a more detailed discussion in this regard and for amle references to the engineering literature dealing with transmission roblems for the Stokes system. If we re-denote the term µ ± u ± in 1.4) as simly u ± and let µ := µ /µ + denote the ratio of the viscosities of the two fluids, we can rewrite the transmission roblem in the form u ± = π ± in Ω ±, T 1 µ) div u ± = 0 in Ω ±, µ u + u = g, ν λ u +, π + ) ν λ u, π ) = f. 1.6) Above, we have also re-denoted the term µ g as simly g, but since we will be interested in considering these roblems for general values of f and g, this is of little consequence. Going one ste further, if we relace π ± with µ ± π ± and f with µ + f in 1.4), we can write a third form of the transmission roblem, u ± = π ± in Ω ±, T 2 µ) div u ± = 0 in Ω ±, u + u = g, ν λ u +, π + ) µ ν λ u, π ) = f. 1.7) 2

5 Since the viscosities µ + and µ are ositive numbers, these changes have no effect on the solvability of these roblems, and so, throughout our work, we will consider the form of the transmission roblem that is most convenient for the articular goals we have in mind. One advantage of these last two descritions comes from analyzing the limiting cases. For examle, if we consider the case when µ << µ +, studying T 1 µ) for µ = 0 yields information about the Regularity roblem R) in Ω, and studying T 2 µ) for µ = 0 yields information about the Neumann roblem N) in Ω +. Similarly, if µ + << µ, analyzing T 1 µ) and T 2 µ) will lead to results for the Regularity roblem R) in Ω + and for the Neumann roblem N) in Ω. Our main results are as follows the reader is referred to the subsequent chaters for the relevant notation emloyed below): Theorem 1.1 Assume that Ω R n, n 2, is a bounded Lischitz domain and set Ω + := Ω, Ω := R n \ Ω. Also, fix µ 0, 1) and λ 1, 1]. Then there exists ε = ε) > 0 such that for each 2n 1) n+1 ε < < 2 + ε, 1.8) the transmission boundary value roblem, concerned with finding two airs of functions u ±, π ± ) in Ω ± satisfying u ± = π ±, div u ± = 0 in Ω ±, and the decay conditions M u ± ), Mπ ± ) L ), u + u = g h 1 ), λ ν u +, π + ) µ λ ν u, π ) = f h ), 1.9) O x 2 n ) as x, if n 3, u x) = 1 µ Ex) ) f dσ + O x 1 ) as x, if n = 2, j u x) = 1 µ je)x) π x) = 1.10) ) f dσ + O x n ) as x, 1 j n, 1.11) O x 1 n ) as x, if n 3, E )x), f dσ + O x 2 ) as x, if n = 2, 1 µ has a unique solution. In addition, there exists C > 0 such that 1.12) M u ± ) L ) + Mπ ± ) L ) C g h 1 ) + C f h ). 1.13) In the revious theorem as well as in the following results, the Hardy sace h ), and its regular version h 1 ), are as defined in 2.97). 3

6 Theorem 1.2 Assume that Ω R n, n 2, is a bounded Lischitz domain. Then there exists ε = ε) > 0 such that for each the interior Dirichlet boundary value roblem 2 ε < < if n = 2, 3, 1.14) 2 ε < < 2n 1) n 3 + ε if n 4, 1.15) u = π, div u = 0 in Ω, M u) L ), u = f L ν + ), 1.16) has a solution, which is unique modulo adding functions which are locally constant in Ω to the ressure term. In addition, there exists a finite constant C > 0 such that M u) L ) C f L ). 1.17) Similar results are valid for the exterior Dirichlet roblem, formulated much as 1.16) with the additional decay conditions { O x 2 n ) as x, if n 3, ux) = Ex) A + O1) as x, if n = 2, { O x 1 n ) as x, if n 3, j ux) = j Ex) A + O x 2 ) as x, if n = 2, { O x 1 n ) as x, if n 3, πx) = E x), A + O x 2 ) as x, if n = 2, 1.18) 1.19) 1.20) for some a riori given constant A R 2. Also, the standard nontangential maximal oerator in 1.17) should be relaced by its truncated version. Here we wish to mention that, while this work was in its final stages of rearation, we have learned that the case of the interior Dirichlet roblem in which the Lischitz domain Ω R n has a connected boundary and n 4 has also been treated by J. Kilty in [54], using a different aroach. The limiting case = has been dealt with by Z. Shen in [82] for Lischitz domains in R 3. In [82], Shen also establishes the well-osedness of the Dirichlet roblem in three-dimensional Lischitz domains with connected boundary for data in the Hölder sace C α ), with 0 < α < α o. Here we give another roof of this result, via integral oerators. In addition, we also treat the case of the Dirichlet roblem for the Stokes system in the case in which the data is from BMO and the solution satisfies Carleson measure estimates. See Theorem 9.16 and Theorem 9.17 for details. Our next result concerns the so-called Regularity roblem, and is a version of the Dirichlet roblem 1.16) corresonding to the case when the boundary data is maximally regular i.e., belonging to boundary Hardy and Sobolev saces of order one). 4

7 Theorem 1.3 Let Ω R n, n 2, be a bounded Lischitz domain. Then there exists ε = ε) > 0 such that for each as in 1.8), the interior Regularity boundary value roblem u = π, div u = 0 in Ω, M u), Mπ) L ), u = f h 1,ν + ), 1.21) has a solution which is unique modulo adding functions which are locally constant in Ω to the ressure. In addition, there exists a finite constant C > 0 such that M u) L ) + Mπ) L ) C f h 1 ). 1.22) Similar results are valid for the exterior Regularity roblem, formulated much as 1.21) with the additional decay conditions 1.18)-1.20). Theorem 1.4 Let Ω R n, n 2, be a bounded Lischitz domain and fix λ 1, 1]. Then there exists ε = ε) > 0 such that for each as in 1.8) the interior Neumann boundary value roblem u = π, div u = 0 in Ω, M u), Mπ) L ), λ ν u, π) = f h ), 1.23) has a solution if and only if ) f Im 1 2 I + K λ : h ) h. 1.24) Ψ λ + Ψ+) λ Moreover, this solution is unique modulo adding to the velocity field functions from Ψ λ Ω). addition, there exists a finite constant C > 0 such that In M u) L ) + Mπ) L ) C f h ). 1.25) Finally, a similar result holds for the exterior domain R n \ Ω after including the decay conditions O x 2 n ) as x, if n 3, ux) = Ex) ) f dσ + O x 1 ) as x, if n = 2, j ux) = j E)x) 1.26) ) f dσ + O x n ) as x, 1 j n, 1.27) O x 1 n ) as x, if n 3, πx) = E )x), f dσ + O x 2 ) as x, if n = ) 5

8 More recisely, a solution to the exterior roblem satisfying the above decay conditions exists if and only if ) f Im 1 2 I + K λ : h h Ψ ) ), 1.29) λ Ψ λ and solutions are unique modulo adding to the velocity field functions from Ψ λ R n \ Ω). Our aroach is based on boundary integral methods, and for each of the roblems listed in Theorems , we are able to reresent the solution in terms of hydrostatic layer otentials. In this strategy, one is led to study the invertibility roerties of certain rincial-value singular integral oerators on Lischitz surfaces. These oerators are of Calderón-Zygmund tye, so their boundedness on Lebesgue and Hardy tye saces follows from known results. The key ingredient in roving the invertibility of these oerators is obtaining bounds from below. We accomlish this by devising some new Rellich tye identities for the Stokes system. The most hysically relevant Neumann-tye boundary condition is the so-called sli condition, corresonding to 1.2) with λ = 1. Interestingly, it is recisely this boundary condition which is most challenging from the oint of view of our analytical treatment. This is because the usefulness of the Rellich tye identities alluded to above is substantially diminished when λ = 1, due to the fact that the quadratic energy form associated with 1.2) when λ = 1 is only semi-ositive definite as oosed to being strictly ositive definite when λ < 1). This difficulty was first encountered by Dahlberg, Fabes, Kenig and Verchota in their work on the L 2 Dirichlet and Neumann roblems for the Stokes and Lamé systems in [23], [34]. As a remedy, these authors have develoed some auxiliary estimates, which they termed boundary Korn inequalities, which were secifically designed to comensate for the lack of coerciveness of the Rellich estimates. In the case of the transmission boundary value roblem for the Stokes system considered here, these Korn inequalities fail to be as useful as they have been in the aforementioned works. This has to do with the very nature of the transmission roblem in which two airs) of solutions u +, π + ) and u, π ), which interact across the Lischitz interface, are considered simultaneously. In this scenario, deriving Korn inequalities for each of them searately is of little value since, in turn, these inequalities cannot be further combined algebraically in order to relate them to the transmission boundary data, i.e., u + u and λ ν u +, π + ) µ λ ν u, π ). 1.30) The technical innovation we develo in order to address this significant issue is to roduce some more elaborate Rellich tye identities which, by design, have Korn-like identities built directly into them. The ushot of this is that working with identities in lace of estimates is amenable to algebraic maniulations which can then fully take advantage of the transmission-like interaction between u +, π + ) and u, π ). All the above considerations are relevant in the treatment of boundary value roblems with L 2 data. As already suggested above, the central role in our treatment is layed by the transmission roblem. Subsequently, we exlain how the Dirichlet/Regularity and Neumann roblems can be viewed as limiting cases of this. To obtain well-osedness results for L -data with 2, following the seminal work of Dahlberg-Kenig [20], [21], we rely on atomic estimates in dimensions n = 2, 3, and on a recent remarkable advance of Z. Shen [83] in dimensions n 4. Shen s original scheme is to start with the L 2 theory, then rove L results for > 2 the critical corresonding to the 6

9 Sobolev exonent in the embedding L 2 1 ) L )) using certain reverse Hölder estimates, and finally interolate. This cannot be directly alied in our setting since the natural range of s for which the L -transmission roblem is solvable is a subset of 1, 2]. We overcome this difficulty by introducing and solving a suitable dual transmission roblem. As is well-known, in the case of the Dirichlet boundary value roblem for the Stokes system, i.e. for u = π, div u = 0 in Ω, u = f, 1.31) the boundary datum f satisfies the necessary comatibility condition ν, f dσ = ) whenever Ω R n is a bounded Lischitz domain. This creates the following technical difficulty when addressing the issue of well-osedness of 1.31) for a bounded Lischitz domain Ω R n when the boundary datum f belongs to the regular) Hardy sace h 1, n 1 at ), n < 1. The latter is the l -san of certain building blocks satisfying suitable suort, size, and smoothness conditions), called regular atoms. Hence, it is natural to seek a solution for 1.31) when f = j λ ja j with λ j ) j l and the a j s regular atoms, as u = j λ j u j, where u j solves 1.31) for the boundary datum a j. However, even though the original datum f satisfies the necessary comatibility condition 1.32), there is no guarantee that each individual atom a j does. We overcome this issue by first addressing the solvability of 1.31) in the case when Ω R n is the unbounded domain lying above the grah of a real-valued) Lischitz function. In this setting, condition 1.32) no longer lays a role. We then develo aroriate localization techniques carried out at the level of singular integral oerators) in order to eventually handle the case of bounded Lischitz domains. This idea influences our overall strategy in dealing with all tyes of boundary conditions for the Stokes system treated in our work. Having develoed a satisfactory theory for the Stokes system with L and atomic) data and nontangential maximal function estimates, we next consider the inhomogeneous Stokes roblem on Besov-Triebel-Lizorkin saces in Lischitz domains. The key idea is to view the former results as limiting/critical cases of the latter, and use interolation. There are, nonetheless, significant difficulties in carrying out this rogram, a fact frequently noted in the literature. For examle, discussing the status of the Poisson roblem for the Stokes system in Lischitz domains, P. Deuring writes on. 3 of [30]: We see that for solutions of the Poisson roblem [for the Dirichlet Lalacian] on Lischitz domains, a rather comlete L -theory is available, whereas for the Stokes system, only a L 2 -theory could be develoed. This, admittedly, was difficult enough, but this still raises the question what to exect if 2. A related oen roblem, osed on. 195 of [28], asks whether for an arbitrary bounded Lischitz domain Ω there holds u π = f L 2 Ω) div u = 0 in Ω u W 1,2 0 Ω), π L 2 Ω) = u W 3/2,2 Ω). 1.33) 7

10 A similar issue is raised in the case of Neumann boundary conditions. In the same setting, Deuring also asks if u = π in Ω div u = 0 in Ω = u W 1/2,2 Ω). 1.34) M u) L 2 ) Here we rovide answers to the above questions and extend revious work in the literature by roving Theorem 1.5 and Theorem 1.6 below. In order to facilitate stating them, we introduce some notation. Let Bα,q R n ) and Fα,q R n ) denote the standard Besov and Triebel-Lizorkin scales of saces in R n cf for more details). Given a Lischitz domain Ω R n and 0 <, q, α R, we set Bα,q Ω) := {u D Ω) : v Bα,q R n ) with v Ω = u}, B,q α,0 Ω) := {u B,q α R n ) : su u Ω}, 1.35) with similar definitions for Fα,q Ω) and F,q α,0 Ω). Also, B,q s ) stands for the Besov class on the Lischitz manifold, obtained by transorting via a artition of unity and ull-back) the standard scale Bs,q R n 1 ). In general, we make no notational distinction between these smoothness saces of scalar-valued functions and their natural counterarts for vector-valued functions.) Finally, for ε > 0 and n 2 let us introduce a two dimensional region R n,ε in the s, 1/)-lane, which deends on the dimension as follows: 1 1, ε) 1 sloe 1 1 -ε, 1) 2 sloe ε,1) 1,1+ε) ε ε 2 1, 1 -ε) , 1 sloe 1 2 ε) O 1 s 2 +ε 1 O 2ε 1 s sloe 1 Figure 1: 2 Figure 2: R n,ε for n = 2 R n,ε for n = 3 8

11 The theorem below deals with the case of Dirichlet boundary conditions. Theorem 1.5 Let Ω be a bounded Lischitz domain in R n, n 2, and assume that n 1 n <, 0 < q, n 1) 1 1) + < s < 1. Consider the following boundary value roblem u π = f B,q s+ 1 2Ω), div u = g B,q s+ 1 1Ω), u B,q Ω), s+ 1 π B,q s+ 1 1Ω), Tr u = h Bs,q ), 1.36) subject to the necessary) comatibility condition ν, h dσ = O O gx) dx, for every comonent O of Ω. 1.37) Then there exists ε = εω) 0, 1] such that 1.36) is well-osed with uniqueness modulo locally constant functions in Ω for the ressure), if the air s, ) belongs to the region R n,ε, deicted above. Furthermore, the solution has an integral reresentation formula in terms of hydrostatic layer otential oerators and satisfies natural estimates. Concretely, there exists a finite, ositive constant C = CΩ,, q, s, n) such that u B,q s+ 1 Ω) + π B,q C f s+ 1 1Ω)/R Ω + B,q + C g B,q + C h B,q s+ 1 2Ω) s+ 1 1Ω) s ). 1.38) Moreover, analogous well-osedness results hold on the Triebel-Lizorkin scale, i.e., for the roblem u π = f F,q,q s+ 1 2Ω), div u = g F s+ 1 1Ω), u F,q Ω), s+ 1 π F,q s+ 1 1Ω), Tr u = g B, s ), 1.39) where the data is, once again, made subject to 1.37). This time, in addition to the revious conditions imosed on the indices, q, it is also assumed that, q <. In the class of Lischitz domains, we conjecture that this result is shar. When C 1, one may take ε = 1. This follows by combining the results in [32] with those of the current work. Theorem 1.5 refines a long list of results in the literature. When is sufficiently smooth, various cases tyically corresonding to Sobolev saces with an integer amount of smoothness) have been dealt with by L. Cattabriga [14], R. Temam [88], Y. Giga [39], W. Varnhorn [92], R. Dautray and J.-L. Lions [25], among others, when is at least of) class C 2. This has been subsequently extended by C. Amrouche and V. Girault [4] to the case when C 1,1 and, further, by G.P. Galdi, C.G. Simader, and H. Sohr [37] when is Lischitz with a small Lischitz constant. There is also a wealth of results related to Theorem 1.5 in the case when Ω is a olygonal domain in R 2, or a olyhedral domain in R 3. A extended account of this field of research can be found in V.A. Kozlov, V.G. Maz ya, and J. Rossmann s monograh [59], which also contains ertinent references to earlier work. Here we also wish to mention the recent work by V. Maz ya and J. Rossmann [65]. Comarison between the regularity results obtained in [59], [65] and our Theorem 1.5 shows that the latter is otimal, at least if n = 2, 3. In the case of the inhomogeneous Neumann roblem we shall rove the following. 9

12 Theorem 1.6 Let Ω be a bounded Lischitz domain in R n, n 2, with connected comlement, and fix n 1 n <, 0 < q, and n 1) 1 1) + < s < 1. Then there exists ε = εω) 0, 1] such that the Poisson roblem for the Stokes system with Neumann boundary condition u π = f, f B,q Ω s+ 1 2,0Ω), div u = 0 in Ω, u B,q Ω), s+ 1 π B,q s+ 1 1Ω), λ ν u, π) f = h B,q s 1 ), 1.40) has a unique solution modulo adding to the velocity functions from Ψ λ Ω)) if the air s, belongs to the region R n,ε described before, and the data f, h) satisfy the necessary comatibility condition Ω f, ψ dx = h, ψ dσ, ψ Ψ λ Ω). 1.41) In addition, the solution normalized so that Ω ux), ψx) dx = 0 for every ψ Ψλ Ω)) satisfies the estimate u B,q s+ 1 Ω) + π B,q C f B,q + C h B,q s+ 1 1Ω) s+ 1 2,0Ω) s 1 ). 1.42) Moreover, an analogous well-osedness result holds for the roblem u π = f, f F,q Ω s+ 1 2,0Ω), div u = 0 in Ω, u F,q Ω), π F,q s+ 1 s+ 1 1Ω), λ ν u, π) f = 1.43) h B, s 1 ), assuming that, q <. Finally, if the condition that the comlement of Ω is connected is droed i.e., Ω R n is an arbitrary Lischitz domains), then roblems 1.40), 1.43) have solutions for data f, h) belonging to a finite co-dimensional subsace of B,q,q s+1/ 2,0 Ω) B,q s 1 ) and Fs+1/ 2,0 Ω) B, s 1 ), resectively, and uniqueness holds u to a finite dimensional sace. Above, λ ν u, π) f should be thought of as a re-normalization of the conormal derivative 1.2) relative to f. See Theorem and the discussion receding it for a more recise formulation. Here we only wish to oint out that when C 1 and λ = 1, corresonding to the so-called sli boundary condition, one can take ε = 1. Theorems are roved by interolating the end-oint cases addressed in Theorems This is done at the level of boundary layer otentials and solutions for the roblems described in Theorems are roduced in a constructive manner, via integral reresentation formulas. 1.2 Consequences of the solvability of the inhomogeneous roblem Here we record several relevant consequences of the well-osedness results from Theorems Denote by G D the Green oerator for the inhomogeneous roblem for the incomressible Stokes system with Dirichlet boundary conditions. That is, formally, if u, π) solve u π = f in Ω, div u = 0 in Ω, Tr u = 0 on, 1.44) 10

13 then G D f := u. 1.45) Corollary 1.7 If Ω is a bounded, Lischitz domain in R n, n 2, then there exists some small number ε = εω) > 0 such the oerators G D : Bα,q Ω) B,q α+2 Ω), 1.46) G D : Fα,q Ω) F,q α+2 Ω), 1.47) are well-defined and bounded whenever 0 < q and the oint with coordinates α 1/ + 2, 1/) belongs to the region R n,ε in Figures 1-3. The two-dimensional region of oints with coordinates α, 1/) for which α 1/+2, 1/) R 3,ε is deicted below: sloe 1 0,1+ ε 3 ) 1 sloe 1 3 ε 2, 1+ ε 2 ) ε,1) 1 3+ε 2, 1+ε 2 ) 1 2 sloe 1 1 ε 2, 1 ε 2 ) -2-2+ε -1-1/2 O α sloe 1 3 Figure 4 Thus, in the setting of a bounded Lischitz domains Ω R 3, the oerators 2 G D : Bα,q Ω) Bα,q Ω), 1.48) 2 G D : Fα,q Ω) Fα,q Ω), 1.49) are bounded whenever 0 < q and the oint with coordinates α, 1/) belongs to the entagonal region from Figure 4. It is interesting to secialize this result to the Triebel-Lizorkin scale with q = 2 and α = 0, in which case one obtains that 2 G D : h Ω) h Ω) boundedly, if Ω R 3 is a bounded Lischitz domain and 1 ε < < 1 for some ε = εω) > ) 11

14 Corresonding to the two-dimensional case we have 2 G D : h Ω) h Ω) boundedly, if Ω R 2 is a bounded Lischitz domain and 2 3 ε < < 1 for some ε = εω) > ) For the Lalace oerator, similar results valid in all sace dimensions) have been established in [63], [64]. This answered in the affirmative a conjecture made by D.-C. Chang, S. Krantz, and E. Stein cf. [15], [16]) regarding the regularity of the harmonic Green otentials on Hardy saces in Lischitz domains. Here we rove the analogue of the Chang-Krantz-Stein conjecture for the Stokes system for arbitrary Lischitz domains in the three dimensional setting. Analogous results are valid for G N, the Green oerator associated with the inhomogeneous Stokes roblem with Neumann boundary conditions. When secialized to the case α = 1 and q = 2, the oerator 1.47) becomes G D : W 1, Ω) W 1, Ω) boundedly, 2n if n+1 ε < < 2n n 1 + ε for some ε = εω) > 0, 1.52) where W s, Ω) stands for the usual L -based Sobolev sace of smoothness s in Ω. This follows from a brief insection of the region in Figures 1-3. As a corollary, for every bounded Lischitz domain Ω R 3 there exists = Ω) > 3 such that the oerator in 1.52) is well-defined and bounded. A similar result is valid for G N. In the case of G D, a result of this tye has first been obtained by R. Brown and Z. Shen in [10] at least if is connected and for Dirichlet boundary conditions). When Ω R 2 is a bounded Lischitz domain, the same tye of conclusion holds for some = Ω) > 4. Let us also single out the following low-dimensional result: Corollary 1.8 Assume that Ω is either a convex olygon in R 2 or a convex olyhedron in R 3. Then G D : L Ω) W 2, Ω) boundedly, whenever 1 < ) Indeed, this follows by interolating between the case 2 3 ε < < 1, contained in 1.51), and the case = 2, which has been dealt with by R.B. Kellogg and J.E. Osborn in [52], when Ω R 2 is a convex olygon, and by M. Dauge in [24] and by V.A. Kozlov, V.G. Maz ya, and C. Schwab in [60] when Ω R 3 is a convex olyhedron. Theorem 1.8 should be comared with the result imlied by the work of V. Kozlov and V. Maz ya in [56], to the effect that G D : L q Ω) L Ω) boundedly, q > 2, rovided Ω R 2 is a bounded convex domain. 1.54) This ortion of our work can be regarded as the natural analogue of the treatment of D. Jerison and C. Kenig of the inhomogeneous Dirichlet roblem for the Lalacian in Sobolev-Besov saces in Lischitz domains from [46]. Here, we are able to extend this to the case of the Stokes system in a Lischitz domain Ω, remove the assumtion that is connected, handle boundary conditions of Neumann tye, and work of more general scales of saces including non locally convex Besov and Triebel-Lizorkin saces). 12

15 We continue by recording the following significant consequence of Theorem 1.5. Related versions for smooth domains have been roved by C. Amrouche and V. Girault in [4], [5], and by V. Girault and P.-A. Raviart in [40]. To state it, introduce Fα,zΩ),q := {u Ω : u Fα,q R n ) su u Ω}, lus a similar definition for Bα,zΩ).,q Corollary 1.9 For every bounded, Lischitz domain Ω in R n, n 2, there exists some small number ε = εω) > 0 such that F,q α,zω; R n ) = { v F,q α,zω; R n ) : div v = 0} B,q α,zω; R n ) = { v B,q α,zω; R n ) : div v = 0} { u Fα,zΩ;,q R n ) : u F,q α 1 Ω)}, 1.55) { u Bα,zΩ;,q R n ) : u B,q α 1 Ω)}, 1.56) where the direct sums are toological, whenever the oint with coordinates α 1/ + 2, 1/) belongs to the region R n,ε in Figures 1-3 and 0 < q. In articular, corresonding to the case when α = 1 in 1.55), rovided W 1, 0 Ω; R n ) = { v W 1, 0 Ω; R n ) : div v = 0} 2n n+1 ε < < 2n n 1 + ε. { u W 1, 0 Ω; R n ) : u L Ω)}, 1.57) Indeed, if w Fα,zΩ;,q R n ) is arbitrary and the air u, π) Fα,zΩ;,q R n ) F,q α 1 Ω) solves 1.39) for f := w F,q α 2 Ω; Rn ), g := 0, and h := 0, then w = u + w u) is the desired decomosition. That sum in the right-hand side of 1.55) is direct is immediate from the uniqueness statement for 1.39). This roves 1.55), and the argument for 1.56) is similar. Finally, 1.57) is a direct consequence of 1.55). We next discuss the analogue of the off-diagonal estimates for the Green oerator associated with the Dirichlet Lalacian in Lischitz domains, established by B.E.J. Dahlberg in [19]. Corollary 1.10 Let Ω R 3 be a bounded Lischitz domain. Then there exists ε = εω) > 0 with the roerty that if then the oerator 1 < < ε and 1 q = ) G D : L Ω) W q 1 Ω) 1.59) is well-defined and bounded. A similar result holds in the case when Ω is a bounded Lischitz domain in R 2, granted that 1.58) is relaced by 1 < < ε and 1 q =

16 To justify this, consider an arbitrary vector field f L Ω) and, by taking the convolution of f extended by zero to R 3 ) with the fundamental solution for the Stokes system in the free sace, construct two functions w W 2 Ω) and ρ W 1 Ω) such that w ρ = f, div w = 0 in Ω, and w W 2 Ω) + ρ W 1 Ω) C f L Ω). Then G Df = w u, where the air u, π) solves u π = 0, div u = 0 in Ω, and Tr u = Tr w on. Note that the comatibility condition 1.37) is automatically satisfied in this case. Also, w W 2 Ω) W q 1 Ω) if 1/q = 1/ 1/3 and, accordingly, Tr w B q,q 1 1/q ). Then Theorem 1.5 imlies that u W q 1 Ω), π Lq Ω), granted that the oint with coordinates 1 1/q, q) belongs to the entagonal region R 3,ε described in 3 Figure 2. A simle analysis shows that this is always the case whenever 2+ε < q < 3 1 ε, for some ε = εω) > 0. The bottom line is that f L Ω) = G D f W q 1 Ω) if 3 2+ε < q < 3 1 ε, 1 q = ) Next, 1.47) with α = 0, q = 2, and classical embeddings give G D : F,2 0 Ω) F,2 1 Ω) if 3 3+ε < < 1, 1 = ) Interolating by the comlex method between 1.60) and 1.61) then yields 1.59) in full, as long as 1 q = and 1 < q < 3 1 ε, a condition imlied by 1.58). Finally, the reasoning for the two-dimensional case is similar. We conclude with a discussion ertaining to the regularity roerties of solutions of ellitic systems in domains with conical singularities. Consider the inhomogeneous Dirichlet roblem LD)u = f in Ω, with zero boundary conditions, 1.62) where LD) is a homogeneous, strongly ellitic, constant coefficient, formally self-adjoint system of order 2m, m N, and Ω R n is a domain with a conical oint at the origin O R n. Assume that f vanishes near O and u is the variational solution of 1.62). As is well-known, u admits a ower-logarithmic asymtotic exansion near O. Somewhat more recisely, near the origin u behaves like a linear combination of terms of the form x λ log x )lj l j l j l)! 0 l l j ) w x l,j x, 1.63) where the exonents λ j C are the eigenvalues of a certain olynomial oerator encil on a domain that is cut out of the unit shere by the cone with vertex at O which is tangent to the boundary of Ω), and the functions w l,j are generalized eigenvectors corresonding to λ j. The oerator encil arises when taking the Mellin transform of LD) and of the oerators intervening in the boundary conditions along this tangent cone. Secific information about the nature of the eigenvalues λ j yields, in turn, regularity roerties for the solution u. For examle, < min j { n } k Re λ j = u W k 14 near O. 1.64)

17 In [57], V. Kozlov and V. Maz ya have shown that, in the above setting, As a consequence of 1.64)-1.65), we then have Re λ j > m n 1)/ ) u W k near O, whenever < n + ε, 1.66) k m + n 1)/2 where ε = εω) > 0. Moreover, in [58], V. Kozlov and V. Maz ya have also shown that 1.65) and, hence, 1.66), is shar in the case when 2m n. When m = 1, i.e., when LD) is a second order oerator, the above analysis gives that u W 1 near O, whenever < 2n + ε. 1.67) n 1 While, strictly seaking, the Stokes system does not fit into this general narrative since it is not ellitic in the sense of I.G. Petrovskii, the same circle of ideas can be adated to this somewhat nonstandard case. See, e.g., the work of V.A. Kozlov, V.G. Maz ya, and C. Schwab in [60] as well as the monograh [59] for the lower dimensional case n = 2, 3). The relevance of the above observation is that is also the critical integrability exonent we have identified in 1.52). Thus, our results are consistent with the redictions of the regularity theory for domains with conical singularities, and are shar when n = 2, 3. While it is not entirely clear whether that is also true when n 4, we conjecture that this is indeed the case. Acknowledgments. We are indebted to R. Brown, M. Costabel, V. Maz ya, S. Nicaise, and Z. Shen for several stimulating discussions and for their interest in this work. Partial suort from the NSF grants DMS and DMS/FRG is also gratefully acknowledged. 2n n 1 2 Smoothness saces and Lischitz domains For a brief review of the Besov and Triebel-Lizorkin scales in the entire Euclidean sace R n, the reader is referred to Grah Lischitz domains We start with a few basic definitions. A grah Lischitz domain Ω R n is simly the domain lying above the grah of a real-valued Lischitz function. That is, Ω := {x = x, x n ) R n 1 R : x n > ϕx )}, where x = x 1,..., x n 1 ), ϕ : R n 1 R is Lischitz, i.e., ϕ exists and belongs to L R n 1 ). We denote by dσ the surface measure on, and by ν the outward unit normal defined a.e. with resect to dσ) on. Hereafter, we will define Ω ± by 2.1) Ω + := Ω and Ω := R n \ Ω. 2.2) 15

18 Next, we define the cones Γ ± κ := {y = y, y n ) R n + : y < ±κy n }, 2.3) and for any x R n, define Γ ± κ x) := x + Γ ± κ. 2.4) In order to introduce the classical non-tangential maximal oerator M, fix some κ = κ) with κ 1 > ϕ L. Then it can be shown that Γ ± κ x) Ω ± for all x. When the value of κ is understood, we will often dro it from the notation and write Γ ± κ x) = Γ ± x). Now, for an arbitrary u : Ω ± R, we set Mu)x) := su { uy) : y Γ ± x)}, x. 2.5) These conical regions also lay a fundamental role in defining non-tangential restrictions to the boundary. Again for u defined in Ω ±, set u x) := lim y x y Γ ± x) uy), for a.e. x. 2.6) Similarly, if, denotes the canonical inner roduct in R n although, later, the same symbol is going to be occasionally used for the airing between a sace and its dual), we set ν ux) := νx), y x lim y Γ ± x) u)y), for a.e. x. 2.7) By L ) we denote the Lebesgue sace of measurable, -th ower integrable functions on, with resect to the surface measure dσ. Next, consider the first-order tangential derivative oerators τjk, acting on a comactly suorted function ψ of class C 1 in a neighborhood of by τjk ψ := ν j k ψ) ν k j ψ), j, k = 1,..., n. 2.8) For every f L 1 loc ), define the functional τ kj f by setting τkj f : C0R 1 n ) ψ f τjk ψ) dσ. 2.9) Thus, if f L 1 loc ) has τ kj f L 1 loc ), the following integration by arts formula holds: f τjk ψ) dσ = For each 1, ), we can then define the Sobolev tye sace τkj f) ψ dσ, ψ C 1 0R n ). 2.10) } L 1 {f ) = L ) : τjk f L ), j, k = 1,..., n. 2.11) 16

19 For each 1 < <, this becomes a Banach sace when equied with the natural norm If we set f L 1 ) := f L ) + tan f := then for each function f L 1 ) n j,k=1 τjk f L ). 2.12) ) ν k τkj f, f 1 j n L 1 ), 2.13) σ-a.e. on. In articular, τjk f = ν j tan f) k ν k tan f) j, j, k = 1,..., n, 2.14) tan f L ) n j,k=1 n 1 τjk f L ) τjn f L ), f L 1 ). 2.15) Furthermore, if 1 <, < are such that 1/ + 1/ = 1 then j=1 τjk f) g dσ = f τkj g) dσ 2.16) for every f L 1 ), g L 1 ). In general, we shall call a first-order differential oerator tangential if it can be written as a variable coefficient) linear combination of the oerators τjk. If Ω R n is the domain lying above the grah of a Lischitz function ϕ : R n 1 R then, for each 1, ), f L 1 ) f, ϕ )) L 1 Rn 1 ), 2.17) with equivalence of norms. As a corollary, we obtain from this that for any bounded Lischitz domain Ω in R n, Li) L 1 ) and C R n ) L 1 ) densely 2.18) whenever 1 < <. For each 1 < <, L 1 ) is a Banach sace, densely embedded into L ). Furthermore, since the maing J : L 1 ) [L ) ] n 1)n 1+ 2, Jf := f, τjk f) 1 j,k n ), 2.19) 17

20 is bounded both from above and below, its image is closed. Now, L 1 ) is isomorhic to the latter sace and, hence, is reflexive. Thus, if for each 1 < <, we set it follows that L 1 ) := L 1 ) ), 1/ + 1/ = 1, 2.20) We can now rove the following result. L 1 ) ) = L 1 ), 1/ + 1/ = ) Corollary 2.1 Let Ω be a Lischitz domain in R n, 1 < < and fix j, k {1,..., n}. Then the oerator τjk : L 1 ) L ) 2.22) extends in a unique) comatible fashion to a bounded, linear maing Proof. For every f L ), set τjk f, g := τjk : L ) L 1 ). 2.23) f τkj g dσ, g L 1 ), 2.24) where 1/ + 1/ = 1. Then the desired conclusion follows from the boundary integration by arts formula 2.16). Corollary 2.2 Assume that Ω is a Lischitz domain in R n and that 1 < <. Then for every f L 1 ) there exist g 0, g jk L ), 1 j, k n not necessarily unique) with the roerty that Furthermore, f = g 0 + n j,k=1 τjk g jk in L 1 ). 2.25) [ f L 1 ) inf g 0 L ) + n j,k=1 g jk L ) where the infimum is taken over all reresentations of f as in 2.25). ], 2.26) 18

21 Proof. Let 1, ) be such that 1/ + 1/ = 1. If f L 1 ) is regarded as a functional f : L 1 ) R, then f J 1 : Im J R is well-defined, linear and bounded where J is as in 2.19) with in lace of ). At this stage, the Hahn-Banach Theorem in conjunction with Riesz s Reresentation Theorem ensure the existence of g 0, g jk L ) such that 2.25)-2.26) hold. Let us also note that, as a simle alication of the one of the standard consequences of the Hahn-Banach theorem, L ) L 1 ) densely, for every 1, ). 2.27) For an unbounded Lischitz domain Ω R n, the homogeneous L -Sobolev sace of order one is defined as L 1 ) := {f L loc ) : τ jk f L ), 1 j, k n}. 2.28) Clearly, for each 1, ), L 1 ) becomes a Banach sace modulo constants when equied with the homogeneous norm f L 1 ) := tanf L ). 2.2 Hardy saces on grah Lischitz surfaces Throughout this section, we shall assume that Ω is as in 2.1), i.e., the unbounded domain in R n lying above the grah of the Lischitz function ϕ : R n 1 R. A surface ball S r x) is any set of the form B r x), with x and 0 < r <. When the center is already secified or of no articular imortance, we simlify the notation by writing S r. For n 1 n < 1, the homogeneous Hardy sace is then defined by { H at ) := f = j λ j a j : a j, o )-atom, λ j ) j l }, 2.29) where the series converges in Li c ), the dual of Li c ), and equied with the usual infimum norm. Here, 1 < o is a fixed arameter and a measurable function a : R is called a, o )-atom if there exists a surface ball S r such that ) su a S r, a L o) r n 1) 1 o 1 and a dσ = ) Given the atomic characterization of Hardy saces in the Euclidean setting, we have f H at ) f, ϕ )) 1 + ϕ ) 2 H at Rn 1 ). 2.31) In articular, this shows that different choices of the arameter o in 2.30) yield the same vector sace and toology on H at ). Let us also recall here the the well-known fact that H at Rn 1 ) = F,2 0 R n 1 n 1 ) if n < 1, 2.32) where F s,q R n 1 ) stands for the homogeneous Triebel-Lizorkin sace in R n 1. See the discussion on. 42 in [36]. For a recise definition, as well as basic roerties of the latter scale see, e.g., [35], [90]. Here we only wish to oint out that, as remarked on. 44 in [36], 19

22 g n 1 Fs,q R n 1 ) j=1 j g F,q s 1 Rn 1 ) 2.33) whenever 0 < <, 0 < q, s R. Recall that, for n 1 n < 1 and ε > 0, a, ε)-molecule adated to a surface ball S r is a function m L 1 ) L 2 ) satisfying i) ii) iii) mx) dσx) = 0, ) 1/2 n 1) S16r mx) 2 dσx) r 1 2 1), S2k+1r\S2kr 1/2 mx) dσx)) 2 2 εk n 1) 2 r) 1 k 2 1), k ) It is well-known that there exists a finite constant κ = κ,, ε) > 0 such that m is a, ε)-molecule = m H at ) and m H at ) κ. 2.35) For uniformity of notation, we find it convenient to define { H H at ) for n 1 n ) := < 1, L ) for > ) Corresonding to one unit more on the smoothness scale we have the regular Hardy sace H 1, n 1 at ), defined for n < 1 as the l -san of regular atoms. More secifically, if [f] denotes the class of f modulo constants, define H 1, at ) := {[f] : f L 1 loc ) and there exist λ i) i l and a i regular, o )-atoms with τjn f = i=1 } λ i τjn a i whenever 1 j n 1, 2.37) where the series converges in Li ). Also, set f H 1, at ) := inf [ λ i ] 1/, where the infimum is taken over all ossible reresentations. Here, if n 1)/n < 1 < o, a function a L o 1 ) is called a regular, o)-atom if there exists a surface ball S r so that ) su a S r, tan a L o) r n 1) 1 o ) In analogy with 2.31), it can be shown that [f] H 1, at ) [f, ϕ ))] F,2 1 R n 1 ). 2.39) Much as before, this shows that different choices of the arameter o in 2.38) yield the same vector sace and toology on H 1, at ). We also set { H 1, H 1 ) := at ) for n 1 n < 1, L 1 ) for > ) An alternative characterization of the quasi-norm in the sace H 1 ) is as follows. 20

23 Lemma 2.3 Let Ω be as in 2.1) and assume that n 1 n < <. Then for each j, k {1,..., n} is a bounded oerator. Furthermore, and, in fact, τjk : H 1 ) H ) 2.41) H 1, at ) = { [f] : f L 1 loc ) and τ jn f H at ) 1 j n 1 }, 2.42) n 1 [f] H 1 ) τjn f H ). 2.43) j=1 Proof. The claim about 2.41) follows straight from definitions when 1 < <, and by analyzing the action of this oerator on atoms when n 1 n < 1. This also yields the right-ointing inequality in 2.43). Now, the oosite inequality is trivial for 1 < <, so there remains to justify it when n 1 n < 1. In this scenario, we note that for every j {1,..., n 1} we have τjn f H at ) 1 + ϕx ) 2 τjn f)x, ϕx )) H at Rn 1 ) 2.44) by 2.32). In concert with 2.33), this ensures that j [fx, ϕx ))] H at Rn 1 ) j [fx, ϕx ))] F,2 0 R n 1 ), τjn f H at ) for every j {1,..., n 1} = fx, ϕx )) F,2 1 R n 1 ). 2.45) If we now recall that, as roved in Proosition 3.4 in [66], it follows that H 1, at Rn 1 ) = F,2 1 R n 1 ) for n 1 n < 1, 2.46) τjn f H at ) for every j {1,..., n 1} = f H1, at ). 2.47) This membershi statement is accomanied by natural estimates and this finishes the roof of 2.43). Now, 2.42) follows from this equivalence. The sace H 1, at ) in 2.37) is defined modulo constants. A realization of this as a sace of genuine functions is as follows. If n 1 n < 1 and 1, ) is such that we set 1 = 1 1 n ) H 1, at ) := {f L ) : f = λ j a j in L ), λ j ) j l, a j regular, o )-atom and equi it with the natural infimum quasi-norm. We then have: j=1 21 }, 2.49)

24 Proosition 2.4 If n 1 n < 1, then the alication is an isomorhism. H 1, at ) f [f] := f + R H1,) 2.50) Proof. The maing 2.50) is clearly one-to-one. The fact that this is also onto follows from the lemma below. Lemma 2.5 Let u be a temered distribution in R n with the roerty that j u H R n ), j = 1,..., n, for some n n+1, n). Then there exists c R such that u c L R n ), where := n n. Proof. For each 1 j n, consider T j to be the convolution integral oerator in R n with the kernel j E )x), where E denotes the fundamental solution for the Lalacian in R n. Classical Calderón-Zygmund theory imlies that at k T j = T j k : H R n ) H R n ), 1 j, k n, are bounded oerators. Furthermore, if n n+1 < <, we have n < <, 2.51) n + 1 j T j = I, the identity oerator on H R n ), 2.52) where reeated indices indicate summation, and if then n n + 1 < < n, 1 := 1 1 n, 1 < <, 2.53) T j : H R n ) L R n ) 2.54) boundedly, by the Fractional Integration Theorem. Next, let u be a temered distribution in R n with the roerty that there exists n n+1, n) such that j u H R n ) for each j = 1,..., n. Set and note that, in the sense of distributions, f j := j u H R n ), j = 1,..., n, 2.55) We claim that, in the sense of distributions, k f j = j f k, j, k = 1,..., n. 2.56) k u T j f j ) = 0, k = 1,..., n. 2.57) 22

25 Once 2.57) has been established, it follows that the temered distribution u T j f j must be a constant c which, in turn, imlies that u c = T j f j L R n ). 2.58) which is what we wanted to rove. Therefore, it remains to justify 2.57). Using notational conventions introduced earlier, we can re-write this in the equivalent form f k = k T j f j ), k = 1,..., n. 2.59) To rove 2.59), based on 2.52) and 2.56), for each k we write as desired. k T j f j ) = T j k f j ) = T j j f k ) = j T j f k ) = f k, 2.60) 1, As a corollary of Proosition 2.4, we obtain that the definition of H at ) is indeendent of the articular choice of o 1, ]. Let us also oint out here that, when used in concert with 2.43), the fact that 2.50) is an isomorhism further entails f eh 1, at ) [f] H 1, at ) n 1 j=1 τjn f H at ), uniformly for f 1, H at ). 2.61) 1, A distinctive feature of H at ) is that this sace is local. This can be justified by analyzing the action of multilication by ψ Li c ) on regular atoms. To this end, it is trivial to check that, < 1 < o, then for each η > 0 there exists C = C, ψ, η,, o ) > 0 such that if n 1 n A regular, o )-atom suorted in a surface ball of radius η = C 1 ψ A is a regular, o )-atom on. 2.62) A more refined version of this result, allowing for atoms suorted in surface balls of arbitrary radii, is as follows. Lemma 2.6 Let Ω be Lischitz domain in R n and assume that n 1 n < 1 and o q, where is as in 2.48). If ψ Li c ) then ψ A is, u to a fixed multilicative constant, a regular, o )-atom on whenever A is a regular, q)-atom on. Proof. To fix ideas, let us assume that su ψ S 1, a surface ball of radius 1, and that ψ L ) + tan ψ L ) 1. Fix a regular, q)-atom A on, i.e. a function A L q 1 ) satisfying su A S r, for some r > 0, and tan A L q ) r n 1) 1 q 1 ). In articular, Poincaré s inequality gives A L q ) Cr tan A L q ) C r 1+n 1) 1 q 1 ). Next, introduce r := min {r, 1} > 0 and note that su ψ A) S r. Going further, write tan ψ A) = ψ tan A + tan ψ)a =: I + II, and use Hölder s inequality in order to estimate 23