STABILITY OF CALDERÓN S INVERSE CONDUCTIVITY PROBLEM IN THE PLANE FOR DISCONTINUOUS CONDUCTIVITIES. Albert Clop. Daniel Faraco.

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1 Inverse Problems and Imaging Volume 4, No., 00, 49 9 doi:0.3934/ipi STABILITY OF CALDERÓN S INVERSE CONDUCTIVITY PROBLEM IN THE PLANE FOR DISCONTINUOUS CONDUCTIVITIES Albert Clop Department of Mathematics and Statistics P.O.Box 35 (MaD), FI-4004 University of Jyväskylä, Finland and Department of Mathematics and Statistics P.O.Box 68, FI-0004 University of Helsinki, Finland albertcp Daniel Faraco Instituto de Ciencias Matemáticas CSIC-UAM-UCM-UC3M and Departamento de Matemáticas Universidad Autónoma de Madrid, Campus de Cantoblanco, 8049-Madrid, Spain Alberto Ruiz Departamento de Matemáticas Universidad Autónoma de Madrid, Campus de Cantoblanco, 8049-Madrid, Spain (Communicated by Giovanni Alessandrini) Abstract. It is proved that, in two dimensions, the Calderón inverse conductivity problem in Lipschitz domains is stable in the L p sense when the conductivities are uniformly bounded in any fractional Sobolev space W α,p α > 0, < p <.. Introduction. Calderón inverse problem, see [], consists in the determination of an isotropic L conductivity coefficient γ on Ω from boundary measurements. These measurements are given by the Dirichlet to Neumann map Λ γ, defined for a function f on Ω as the Neumann value Λ γ (f) = γ ν u, where u is the solution of the Dirichlet boundary value problem { (γ u) = 0 () u Ω = f 000 Mathematics Subject Classification. 35R30, 35J5, 30C6. Key words and phrases. Calderón s problem, stability, Inverse problem. The first author is partially supported by projects Conformal Structures and Dynamics, GALA (contract no ), 009-SGR-40 (Generalitat de Catalunya) and MTM (Spain). The second and third authors are partially supported by projects MTM C0-0 and MTM of Ministerio de Educación y Ciencia, Gobierno de España and by the project SIMUMAT from CAM. 49 c 00 American Institute of Mathematical Sciences

2 50 Albert Clop, Daniel Faraco and Alberto Ruiz and ν denotes the outer normal derivative. For general domain and conductivities u has no meaning, the Dirichlet to Neumann map where the pointwise definition γ ν () Λ γ : H / ( Ω) H / ( Ω) can be defined by (3) Λ γ (f), ϕ 0 = Ω γ u ϕ where ϕ W, (Ω) is a function such that ϕ Ω = ϕ 0 in the sense of traces. Since the foundational work of Calderón, research on this question has been very intense but it is not until 006 when, by means of quasiconformal mappings, K. Astala and L. Päivärinta in [], see also [], were able to establish the injectivity of the map γ Λ γ for an arbitrary L function bounded away from zero. Previous planar results were obtained in [35], [45] and [0]. In higher dimensions, the known results on uniqueness require some extra a priori regularity on γ (basically some control on 3 derivatives of γ, see [44], [7], [38] and [9].) A relevant question (specially in applications and in the development of recovery algorithms, see [30] and [6]) is the stability of the inverse problem, that is, the continuity of the inverse map Λ γ γ. For dimension n >, the known results are due to Alessandrini [4], [5]. There the author proved stability under the extra assumption γ W,. In the planar case, n =, the situation is different. Liu proved stability for conductivities in W,p with p > in [3]. In [3], stability was obtained when γ C +α with α > 0. Recently, Barceló, Faraco and Ruiz [4] obtained stability under the weaker assumption γ C α, 0 < α <. Precisely, they prove that for any two conductivities γ, γ on a Lipschitz domain Ω, with a priori bounds K γ i K, K and γ i C α Λ 0, the following estimate holds: γ γ L (Ω) V ( Λ γ Λ γ H / ( Ω) H / ( Ω)) with V (t) = C log( t ) a. Here C, a > 0 depend only on K, α and Λ 0, and f(x) f(y) f C α = sup x y x y α is the seminorm of the class C α of Hölder continuous functions. An example, due to Alessandrini [4], shows that in absence of continuity L estimates do not hold. Namely, if we denote by B r0 = {x R, x < r 0 } the ball centered at the origin with radius r 0, take Ω = B the unit ball in R, γ = and γ = + χ Br0, then γ γ L (Ω) =, but Λ γ Λ γ H H r 0 0 as r 0 0. A closer look to the previous example shows that lim r0 0 γ γ L (Ω) = 0. Therefore one could conjecture that, in absence of continuity, average stability (in the L sense) might hold. However, it is well known that some control on the oscillation of γ is needed to obtain stability. Namely, let γ be defined in the unit square and extended periodically, and denote γ j (x) = γ(jx). Then the sequence {γ j } j=

3 Stability of Calderón s Problem 5 G-converges to a matrix γ 0 (see for example [46] for the notion of G-convergence). Since G-convergence implies the convergence of the fluxes [46, Proposition 9], we get that if u j, u 0 solve the corresponding Dirichlet problems for a fixed function f H ( Ω), { (γ j u j ) = 0 (4) u j Ω = f then the fluxes satisfy that γ j u j γ u. Thus, by (3) (5) lim j,j (Λ γ j Λ γj )(f), ϕ 0 = 0 for each f, ϕ 0 H. However, γ j has no convergent subsequence in L. Notice that γ j can be chosen even being C, so the problem here is not so much a matter of regularity but rather a control on the oscillation. In [6] it is provided a specific choice of γ where the pointwise convergence (5) is strengthened to convergence in the operator norm H H. In this paper we prove that L stability holds if we prescribe a bound of γ in any fractional Sobolev space W α,. By the relation with Besov spaces this could be interpreted as controlling the average oscillation of the function. Thus average control on the oscillation of the coefficients yields average stability of the inverse problem. Theorem.. Let Ω be a Lipschitz domain in the plane. Let γ = γ, γ be two planar conductivities in Ω satisfying (I) Ellipticity: K γ(x) K. (II) Sobolev regularity: γ i W α,p (Ω) with α > 0, < p <, and γ i W α,p (Ω) Γ 0. Let α < min{α, } and ρ = Λ γ Λ γ H / ( Ω) H / ( Ω). Then there exists two constants c(k, p), C(K, α, p) > 0, such that (6) γ γ L (Ω) C( + Γ 0) log(/ρ) c α whenever ρ < /. The theorem is specially interesting for α 0. Then we are close to obtaining stability for conductivities in L and we allow all sort of wild discontinuities. Arguing by interpolation one can also obtain L p stability estimates. Concerning the logarithmic modulus of continuity, the arguments of Mandache [34] can be adapted to the L setting. Namely we can consider the same set of conductivities with the obvious replacement of the C m function by a normalized W α, function. The argument shows the existence of two conductivities such that γ γ L (D) ǫ, γ i W α,p (Ω) Γ 0, but (7) γ γ L (D) C log(ρ) 3(+α) α Here C is a constant depending on all the parameters. Notice that the power is better than in the L setting but still the modulus of continuity is far from being satisfactory. For ρ > it is trivial to obtain Lipschitz stability since the conductivities are uniformly in L..

4 5 Albert Clop, Daniel Faraco and Alberto Ruiz In our way to prove Theorem. we have dealt with several questions related to quasiconformal mappings of independent interest. More precisely, we have needed to understand how quasiconformal mappings interact with fractional Sobolev spaces. In particular we analyze the regularity of Beltrami equations with Sobolev bounds on the coefficients which has been a recent topic of interest in the theory. See [3, 4] where the case µ W,p is investigated in relation with the size of removable sets. We prove the following regularity result. Theorem.. Let α (0, ), and suppose that µ, ν W α, (C) are Beltrami coefficients, compactly supported in D, such that µ(z) + ν(z) K K +. at almost every z D. Let φ : C C be the only homeomorphism satisfying φ = µ φ + ν φ and φ(z) z = O(/z) as z. Then, φ(z) z belongs to W +θα, (C) for every θ (0, K ), and ) D +θα (φ z) L (C) C K ( µ θ W α, (C) + ν θ W α, (C) for some constant C K depending only on K. Many corollaries can be obtained from this theorem by interpolation. An interesting case is for example what do you obtain if µ is a function of bounded variation. We have contented ourselves with the L setting but similar results hold in L p. As a consequence of this theorem, we obtain the corresponding regularity of the complex geometric optics solutions. The other crucial ingredient in our proof is the regularity of µ ψ where ψ is a normalized quasiconformal mapping. It is well known that quasiconformal mappings preserve BMO and Ẇ, (see [39]). Then an interpolation argument is used in [40] to prove that the same happens with Ẇ α, α, 0 < α <. For more general fractional spaces, we prove the following statement: (8) µ W α, µ ψ W β,, for every β < α K which suffices for our purposes. The proof relies on the precise bounds for the powers that Jacobians of quasiconformal mappings to be Muckenhoupt weights obtained in [0]. The Lipschitz regularity of the domain Ω is used to reduce the problem to the unit disk D. This reduction relies on two facts. First, any Lipschitz domain Ω is an extension domain for fractional Sobolev spaces. Secondly, the characteristic function χ Ω belongs to W α, (C) for any α <. Indeed, this is responsible also of the constraint α < at Theorem.. In fact, a stability result holds as well if Ω is any simply connected extension domain. To see this, recall that planar simply connected extension domains Ω are quasidisks ([6]), that is, Ω = φ(d) where φ : C C is quasiconformal. Therefore, for instance by our results in Section 4, χ Ω = χ D φ belongs to some space W α,, and then use Theorem.. The rest of the paper is organized as follows. In Section we recall previous facts from [, 4] which will be needed in the present paper, and describe the strategy of our proof. In Section 3 we reduce the problem to conductivities γ such that γ W α, 0 (D). In Section 4 we study the interaction between quasiconformal mappings and fractional Sobolev spaces. Finally in Section 5 we prove the subexponential

5 Stability of Calderón s Problem 53 growth of the complex geometric optic solutions and in Section 6 we prove the theorem. In closing we remark several issues raised by our work. The first one is to improve the logarithmic character of the stability. It was proved by Alesssandrini and Vesella that often a logarithmic estimate yields Lipschitz stability for some finite dimensional spaces of conductivities. However, to achieve the desired estimates in our setting seems to require a more subtle understanding of the Beltrami equation and we leave it for the future. It will also be desirable to obtain L p estimates in terms of W α,p with constants independent of p, so that the C α situation in [4] could be understood as a limit of this paper. Finally, from the quasiconformal point of view, there seems to be room for improvement in our estimates specially concerning the composition which is far from being optimal when α ր, since Ẇ, is invariant under composition with quasiconformal maps. This will also be the issue for further investigations. Notation. For any multiindex α = (α, α ), we write α = x α y α and α = α + α. The complex derivatives are then z = = z = ( x + i y ) z = = z = ( x i y ) where z = x + iy. For a mapping φ : Ω C, φ(z) = Dφ(z) = φ(z) + φ(z) is the operator norm of the differential matrix Dφ, and J(z, φ) = φ(z) φ(z) is the Jacobian determinant. The fractional derivatives D α f are defined in (), along the work we denote the ordinary differential by Df but, when this notation is not clear, we will denote it by f. Given a Banach space X we denote the operator norm of T : X X by T X. By C or a we denote constants which may change at each occurrence. We will indicate, when necessary, the dependence of the constants on parameters K, Γ, etc, by writing C = C(K, Γ,...). This tracking of the constants is essential for stability results. By X Y we mean that there exists a harmless constant C such that X C Y.. Scheme of the proof. We will follow the strategy of [4]. This work focuses on the approach based on the Beltrami equation initiated in []. The starting point is the answer to Calderón conjecture in the plane obtained by Astala and Päivärinta. Theorem. (Astala-Päivärinta). Let Ω R be a bounded simply connected domain, and let γ i L (Ω), i =,. Suppose that there exist a constant K > such that K γ i K. If Λ γ = Λ γ then γ = γ. In other words, the mapping γ Λ γ is injective. We recall the basic elements from [] needed in the sequel, also the strategies for uniqueness and stability, and what we will need in the current paper. Equivalence between Beltrami and conductivity equation: Let D be the unit disc. If a function u is γ-harmonic in D, then there exists another function v, called its γ-harmonic conjugate (and actually γ -harmonic in Ω), unique modulo constants, such that f = u + iv satisfies the R-linear Beltrami type equation (9) f = µ f

6 54 Albert Clop, Daniel Faraco and Alberto Ruiz with (0) µ = γ + γ R. Then if K is the ellipticity constant of γ we denote by κ = K K +. It is easy to see that µ κ and thus the Beltrami equation is elliptic if and only if the conductivity equation is elliptic. Moreover, for x ( K, K), the function F(x) = x +x satisfies +K F (x) K +K. Thus, it also follows that C γ W α,p (Ω) µ W α,p (Ω) C γ W α,p (Ω), where the constant C only depends on K (see Lemma 3.). Therefore, bounds in terms of µ and γ are equivalent. We can argue as well in the reverse direction. If f W, loc (D) satisfies (9) for real µ with µ κ, then we can write f = u + iv where u and v satisfy div (γ u) = 0 and div ( γ v ) = 0. Thus, it is equivalent to determine either γ or µ, and throughout the paper we will work with either of them interchangeably. As for holomorphic functions, u and v are related by the corresponding Hilbert transform H µ : H ( D) H ( D) defined as H µ (u D ) = v D for real functions, and R-linearly extended to C-valued functions by setting H µ (iu) = i H µ (u). Since T H µ = Λ γ it follows [, Proposition.7] that H µ, H µ and Λ γ are uniquely determined by Λ γ. Accordingly in [4, Proposition.] it is shown that H µ H µ Λ γ Λ γ, with respect to the corresponding operator norms. Existence of complex geometric optics solutions, scattering transform and k equations: The theory of quasiconformal mappings and Beltrami operators allows to combine in an efficient way ideas from complex analysis, singular integral operators and degree arguments to prove the existence of complex geometric optics solutions with no assumptions on the coefficients. Theorem.. Let κ (0, ), and let µ be a real Beltrami coefficient satisfying µ K K+ χ D. For every k C and p (, + κ ) the equation f = µ f admits a unique solution f µ W,p loc (C) of the form () f µ (z) = e ikz M µ (z, k) such that M µ (z, k) = O(/z) as z. Moreover, ( ) M µ Re > 0 and f µ (z, 0) =. M µ

7 Stability of Calderón s Problem 55 In this context, the proper definition of scattering transform of µ (or of γ) is () τ µ (k) = i ( ) e ikz (f µ (z) f µ (z)) da(z). 4π z D Alternatively the scattering transform is given by the asymptotics of the scattering solutions. Namely, (3) τ µ (k) = lim z z(m µ(z, k) M µ (z, k)) The complex geometric optics solutions {u γ, ũ γ } to the divergence type equation () are then obtained from the corresponding ones from the Beltrami equation by u γ = Re(f µ ) + i Im(f µ ) ũ γ = Im(f µ ) + i Re(f µ ), and they uniquely determine the pair {f µ, f µ } (and viceversa) in a stable way. We consider u γ as a function of (z, k). In the z plane, u γ satisfies the complex γ-harmonic equation, div(γ u γ ) = 0. As a function of k, u γ is a solution to the following -type equation u γ (4) k (z, k) = i τ µ(k)u(z, k). Let us emphasize that τ µ (k) is independent of z. Strategy for uniqueness: Let γ, γ be two conductivities. In [], the strategy for uniqueness is divided in the following steps: (i): Reduction to D. (ii): If Λ γ = Λ γ, then τ µ = τ µ. (iii): Step (ii) and (4) imply that u γ = u γ. (iv): Finally, condition u γ = u γ is equivalent to Du γ = Du γ, which holds as well if and only if γ = γ. The first step is relatively easy since there is no regularity of γ to preserve and thus one can extend by 0 in D \ Ω. Second step is dealt with in [, Proposition 6.]. It is shown that H µ = H µ implies f µ (z, k) = f µ (z, k) for all k C and z >. As a consequence (ii) follows from the characterization of τ µ (3). The step (iii) is more complex because uniqueness results and a priori estimates for pseudoanalytic equations in C like (4) only hold if either the coefficients or the solutions decay fast enough at. Unfortunately the needed decay properties for τ seem to require roughly one derivative for γ. However in [] it is shown that in the measurable setting at least one can obtain subexponential decay for the solutions. That is, (5) u γ (z, k) = e ik(z+ǫµ(z,k)) for some function ǫ = ǫ µ (z, k) satisfying lim ǫ µ(z, k) L k (C) = 0. This would not be enough if we would consider equation (4) for a single z. However, in [] it is used that u(z, k) solves an equation for each z. Further, one has asymptotic estimates for u both in the k (as above) and z variables. Then, a clever topological argument in both variables shows that, with these estimates, τ µ determines the solution to (4).

8 56 Albert Clop, Daniel Faraco and Alberto Ruiz Strategy for stability: In order to obtain stability, the natural idea is to try to quantify in an uniform way the arguments for uniqueness. This was done in [4] for C α conductivities. Let us recall the argument and specially the results which did not require regularity of γ and would be instrumental for the current work. Let ρ = Λ γ Λ γ. First one reduces to the unit disk by an argument which involves the Whitney extension operator, the weak formulation (3) and a result of Brown about recovering continuous conductivities at the boundary ([8]). Next we investigate the relation between the corresponding scattering transforms. Theorem.3 (Stability of the scattering transforms). Let γ, γ be conductivities in D, with K γ i K, and denote µ i = γi +γ i. Then, for every k C it holds that (6) τ µ (k) τ µ (k) c e c k ρ. where the constant c depends only on K. The estimate is just pointwise but on the positive side it holds for L conductivities. In [4, Theorem 4.6] there is an explicit formula for the difference of scattering transforms which might be of independent interest. Next we state a result that is implicitly proved in [4, Theorem 5.]. There it is stated as a property of solutions to regular conductivities. However, in the proof the regularity is only used to obtain the decay in the k variable. Because of this, here we state it separately as condition (7). Theorem.4 (A priori estimates in terms of scattering transform). Let K and γ, γ be conductivities on D, with K γ i K. Let u γj (z, k) = e ik(z+ǫµ j (z,k)), denote, as in (5), the complex geometric optics solutions to (). Let us assume that there exist positive constants α, B such that for each z, k C, (7) ǫ µi (z, k) B k α. Then it follows that: A: There exists new constants b = b(k), C = C(K, B), such that for every z C there exists w C satisfying: (a) z w CB log ρ bα, where ρ = Λ γ Λ γ. (b) u γ (z, k) = u γ (w, k). B: For each k C, there exists new constants b = b(k) and C = C(k, K) such that (8) u γ (z, k) u γ (z, k) L (D,dA(z)) CB K log(ρ) bα. Proof. The proof of A follows from [4, Proposition 5.] and [4, Proposition 5.3]. Let us prove B. Given z C, let w C be given by part A. Then u γ (z, k) u γ (z, k) = u γ (z, k) u γ (w, k). By the Hölder continuity of K-quasiregular mappings, together with (a), we get u γ (z, k) u γ (z, k) C(k, K) z w K C(k, K)C K B K log bα K ρ and the desired estimate follows after renaming the constants.

9 Stability of Calderón s Problem 57 Unlike in the uniqueness arguments, estimating D(u γ u γ ) in terms of u γ u γ is more delicate in the stability setting, since functions do not control their derivatives in general. This is solved in [4], under Hölder regularity. Theorem.5 (Schauder estimates). Let γ i, i =, be conductivities on D, such that K γ i K and γ Cα (D) Γ 0. As always, denote µ i = γi +γ i, and let f µi (z, k) be the corresponding complex geometric optics solutions to (9). Then. For each k C there exists a constant C = C(k) > 0 with (9) f µ (, k) f µ (, k) C +α (D) C(k).. The jacobian determinant of f µi (z, k) has a positive lower bound J(z, f µi (, k)) C(K, k, Γ 0 ) > 0., α}. For this, it is used there that characteristic functions of Lipschitz domains belong to W β,q (C) whenever βq <. Then we proceed by investigating the regularity of solutions of Beltrami equations with coefficients in fractional Sobolev spaces in order to obtain an estimate like (9), with the C +α norm replaced by the sharp Sobolev norm attainable under our assumption on the Beltrami coefficient (see Theorem 4.5). It is also needed here to understand how composition with quasiconformal mappings affects fractional Sobolev spaces. As far as we know, the estimates here are new and of their own interest. Afterwards we prove that our Sobolev assumption on µ suffices to get the uniform subexponential growth of the geometric optics solutions needed in condition (7) in Theorem.4 (this is done in Section 5, see Theorem 5.6). In fact we obtain a very clean expression for the precise growth, achieving that the exponent depends linearly on α. Finally, in Section 6 we do the interpolation argument. Here we do not have enough regularity to control W, norms and here is where one sees why we need to be happy with the control on µ µ L (D). Also we do not have a pointwise lower bound for the corresponding Jacobians which causes also difficulties. Now, to finish the proof of stability for Hölder continuous conductivities, just note that an interpolation argument between L and C +α gives Lipschitz bounds for Df µi. Thus, by µ = f and the second statement above, one obtains f L stability for µ µ. The corresponding result for γ γ comes due to (0). Strategy for stability under Sobolev regularity. In the current work we will try to push the previous strategy to obtain L stability. The previous analysis shows that we can rely on many of the results from [, 4]. In particular, we only have to prove that τ µ µ is continuous. For this, we start by reducing the problem in Section 3. We replace the assumption γ i W α,p (Ω) by γ i W β, 0 (D), where 0 < β < min{ 3. Fractional Sobolev spaces and reduction to µ W α, 0 (D). 3.. On fractional Sobolev spaces. Here and in the rest of the section we consider p <. Following [, p.], for any domain Ω, we denote by Ẇ,p (Ω) the class of distributions f with L p (Ω) distributional derivatives of first order. This means that for any constant coefficients first order differential operator D there exists an L p (Ω) function Df such that f, Dϕ = Df ϕ. Ω

10 58 Albert Clop, Daniel Faraco and Alberto Ruiz whenever ϕ C is compactly supported inside of Ω. We also denote W,p (Ω) = L p (Ω) Ẇ,p (Ω). Similarly one can define the Sobolev spaces W m,p (Ω) and Ẇ m,p (Ω) of general integer order m. These are Banach spaces with the norms f W m,p (Ω) = α f Lp (Ω) and f Ẇ m,p (Ω) = α f Lp (Ω). α m α =m Let us introduce for general domains Ω and any real number 0 < α < the complex interpolation space (0) W α,p (Ω) = [L p (Ω), W,p (Ω)] α, and similarly for the homogeneous case Ẇ α,p (Ω) = [L p (Ω), Ẇ,p (Ω)] α. Then the closure of C0 (Ω) (C functions with compact support contained in Ω) in W α,p (Ω) is denoted by W α,p 0 (Ω). For simplicity, H (Ω) = W, (Ω), H0(Ω) = W, 0 (Ω) and H ( Ω) = H (Ω)/H0 (Ω). It comes from the work of Calderón (see [, p.7] or [4]) that every Lipschitz domain Ω is an extension domain. Given Ω Ω, we denote the corresponding extension operator by E, E : W m,p (Ω) W m,p 0 (Ω ) When Ω is an extension domain, an interpolation argument (see [, p.]) shows that W α,p (Ω) coincides with the space of restrictions to Ω of functions in W α,p (C). That is, to each function u W α,p (Ω) one can associate a function ũ W α,p (C) such that ũ Ω = u and ũ W α,p (C) C u W α,p (Ω). We have chosen just one way to introduce the fractional Sobolev spaces. Next, we discuss alternative characterizations and further properties of these spaces needed in the rest of the paper. Two good sources for the basics of this theory are [, Chapter 7], [4, Chapter 4]. Fourier side. By denoting e k (z) = e ikz+ikz, the Fourier transform can be defined as ˆf(k) = e k (z)f(z)da(z). C Then, if f W α,p (C) we introduce the fractional derivative of order α as () D α f(ξ) = ξ α ˆf(ξ). When p =, it is easy to see that { } W α, (C) = f L (C); ( + ξ ) α f(ξ) L (C), and that this agrees with the space of Bessel potentials W α, (C) = G α L (C) = { f = G α g; g L (C) }, where G α is the Bessel kernel [, p.0]. Similarly Ẇ α, (C) = I α L (C) for the Riesz kernels I α. If p, the situation is more complicated but it can be shown that { ) } W α,p (C) = f L p (C); (( + ξ ) α f(ξ) L p (C).

11 Stability of Calderón s Problem 59 Integral modulus of continuity. We define the L p -difference of a function f by () ω p (f)(y) = f( + y) f( ) L p (C). (see [4, Chapter V]). Then the Besov spaces Bα p,q (C) are defined by Bα p,q (C) = {f L p (C) : f q B = ω α p,q p (f)(y) q y (n+αq) dy < }. C There are many relations between Besov and fractional Sobolev spaces. We will need the following two facts, (3) B, α = W α,, W α,p B p, α (p < ). For a proof see [, Chapter 7] or [4, Chapter V]. Generalized Leibnitz Rule. The following result is shown in [9]. See also [7] and [47]. Lemma 3.. Let f, g C 0 (C). (a) Let α, α [0, α] [0, ) be such that α + α = α. Let also p, p (, ) satisfy p + p = p. Then D α (fg) f D α (g) g D α (f) L p C D α (f) L p D α (g) L p, for some constant C = C(α, α, α, p, p, p) > 0. (b) If 0 < α, p (, ] and p (, ) satisfy p + p = p, then D α (f g) L p C (Df) (g) L p D α g L p, for some constant C = C(α, p, p, p). (c) If 0 < α < and < p < then D α (fg) f D α (g) g D α (f) L p C D α (f) L p g L, for some constant C = C(α, p) > 0. Remark. From property (a) and (c) it follows the generalized Leibnitz rule (4) D α (f g) L p C 0 D α f L p g L p + D α g L p 3 f L p 4 whenever < p, p 3 < and p, p 4 and p = p + p = p 3 + p 4. Moreover, we can localize the support in (4) in the following way. Let us assume suppf D, then (5) D α (f g) Lp (D) C 0 f W α,p (D) g L p + D α g L p 3 f L p 4(D) The key point is to use a cutoff function φ with φ = on D and supported on D, and by using (a) and (c) above, we can write D α (f) Lp (D) = D α (fφ) Lp (D) C D α φ L p f L p + D α φ L (D) f L p (D) + φd α f L p. We need to take p > p. This can be achieved by using Sobolev embedding to finally obtain that W α,p (D) L p (D), with p p αp (6) D α f L p C f W α,p (D).

12 60 Albert Clop, Daniel Faraco and Alberto Ruiz Pointwise Inequalities. Lemma 3.. [ [43]] If f W α,p (C), α > 0, < p <, then for each 0 λ α there exists a function g = g λ L p λ (C), p λ = such that p (α λ)p (7) f(z) f(w) z w λ (g(z) + g(w)) for almost every z, w C. Furthermore, we have that for some constant C > 0. g L p λ C f W α,p (C), 3.. Reduction to p =. This reduction relies on the fact that µ L (C) W α,p (C) and the following interpolation Lemma. Lemma 3.3. Let f W α0,p0 W α,p, where < p 0, p <, 0 α 0, α, and θ (0, ). Then, f W α,p f θ W α 0,p 0 f θ W α,p where α = θ α 0 + ( θ)α and p = θ + θ. p 0 p Furthermore, if either p 0 = or p =, then the above inequality holds true by replacing W αi,pi by the Riesz potentials space I αi BMO. Proof. It is well known that the complex interpolation method gives [W α0,p0, W α,p ] θ = W α,p whenever < p < (for the proof of this, see for instance [49]). For p =, the same result holds true if we replace W α, by the space of Riesz potentials I α BMO of BMO functions (for this, see [40]). Let µ be a compactly supported Beltrami coefficient. Then, it belongs both to L (C) and L (C). If we also assume that µ W α,p (C) for some α, p, then we can use the above interpolation to see that µ W β,q (C), for any < q < and some 0 < β < α. We are particularly interested in q =. Lemma 3.4. Suppose that µ W α,p (Ω) L (Ω) for some p > and 0 < α <. Then, For any 0 θ, For any 0 θ, One always has µ W αθ, p θ (Ω) µ θ L (Ω) µ θ W α,p (Ω). µ W θα, where β = αp and p = min{p, p ( θ)p+θ (Ω) µ θ L (Ω) µ θ W α,p (Ω). µ W β, (Ω) C(K, p) µ p / W α,p (Ω), p p }. Proof. The first inequality comes easily interpolating between L (Ω) and W α,p (Ω) (the L norm can even be replaced by the BMO norm, which is smaller, see [40] for more details). For the second, simply notice that compactly supported Beltrami coefficients belong to all L p (Ω) spaces, p >, so one can do the same between L +ε (Ω) (ε as small as desired) and W α,p (Ω). The last statement is obtained by letting θ = p above.

13 Stability of Calderón s Problem Reduction to Ω = D and µ W α,p (D), µ K K+ χ D. The proof of the following lemma relies on the fact that characteristic functions of Lipschitz open sets belong to W α, for each α <. Theorem 3.5. Let Ω be a Lipschitz domain, strictly included in D. Let µ W α, (Ω). Define { µ Ω µ = 0 C \ Ω. Then, µ W β, 0 (C) for β < min{α, } and µ W β, (C) C µ W α, (C). Analogous results can be stated for the extensions by of γ i. Proof. Since Ω is an extension domain, there is an extension µ 0 of µ belonging to W α, (C). Of course, such extension µ 0 need not be supported in Ω any more. Now µ can be introduced as the pointwise multiplication µ = χ Ω µ 0. By virtue Lemma 3. it is enough to study the smoothness of the characteristic function χ Ω. A way to see this is to recall that fractional Sobolev spaces are invariant under composition with bilipschitz maps [50]. Now, the characteristic function of the half plane belongs to W α,p loc (C) whenever αp <. Therefore, by a partition of unity argument, we get that χ Ω W α,p (C) when αp <. The proof is concluded. Now we need to compare the original Dirichlet-to-Neumann maps with the Dirichletto-Neumann maps of the extensions. Lemma 3.6. Let Ω be a domain strictly included in D. Let γ, γ L (Ω) be conductivities in Ω. Further, assume that K γ i(z) K for almost every z Ω. Let γ i denote the corresponding extensions by to all of C. Then, where ρ = Λ γ Λ γ H ( Ω) H ( Ω). Λ γ Λ γ H ( D) H ( D) C ρ, Proof. We follow the ideas of [4, Theorem 6.], although the stability result from [8] is not needed in our situation. Let ϕ 0 H ( D). Let ũ j H (D) be the solution to { ( γ j ũ j ) = 0 in D ũ j = ϕ 0 in D. Let also u be defined by { (γ u ) = 0 u = ũ in Ω in Ω.

14 6 Albert Clop, Daniel Faraco and Alberto Ruiz Define now ṽ = u χ Ω + ũ χ D\Ω. As in [4], we first control ũ ṽ in terms of ρ. To do this, (ṽ ũ ) c γ (ṽ ũ ) (ṽ ũ ) D D = c γ ṽ (ṽ ũ ) D because ṽ ũ H0 (D) and the γ -harmonicity of ũ in D. By adding and subtracting D γ ũ (ṽ ũ ), and using that γ = γ = off Ω, the right hand side above is bounded by a constant times γ ũ (ṽ ũ ) + (γ ũ γ u ) (ṽ ũ ). D Ω Here the first term vanishes because ũ is γ -harmonic on D and ṽ ũ H0 (D). For the second, we observe that ũ is γ -harmonic in Ω, u is γ -harmonic in Ω, and u ũ H0 (Ω). Thus, (γ ũ γ u ) (ṽ ũ ) = (Λγ Λ γ )(ũ Ω ), (ṽ ũ ) Ω Ω ρ ũ H ( Ω) ṽ ũ H ( Ω) ρ ũ L (Ω) (ṽ ũ ) L (Ω) Summarizing, we get ( ) (ṽ ũ ) c ρ ũ L (Ω) c ρ ũ L (D) (8) D c ρ ϕ 0 H. ( D) We will use this to compare the Dirichlet-to-Neumann maps at D. If ψ 0 H ( D) is any testing function, and ψ is any H (D) extension, (9) (Λ γ Λ γ )(ϕ 0 ), ψ 0 = ( γ ũ γ ũ ) ψ. We will divide the bound of this quantity in two steps. For the first, ( γ ũ (γ χ Ω + γ χ D\Ω ) ṽ ) ψ = (Λγ Λ γ )(ũ Ω ), ψ Ω D which is bounded by ρ ũ Ω H ( Ω) ψ Ω H ( Ω) ρ ũ L (Ω) ψ L (Ω) D ρ ũ L (D) ψ L (D) ρ ϕ 0 H ( D) ψ 0 H ( D). We are left with D ( (γ χ Ω + γ χ D\Ω ) ṽ γ ũ ) ψ which is equal to Ω γ (ṽ ũ ) ψ + D\Ω (ṽ ũ ) ψ

15 Stability of Calderón s Problem 63 which in turn is controlled, using (8), by a multiple of (ṽ ũ ) ψ (ṽ ũ ) L (D) ψ L (D) D c ρ ϕ H ( D) ψ 0 H ( D). This gives for (9) that the difference of Dirichlet-to-Neumann maps satisfies as desired. (Λ γ Λ γ )(ϕ 0 ), ψ 0 c ρ ϕ H ( D) ψ 0 H ( D) Remark. The trivial extension of the conductivities by simplifies the arguments but has the price of losing regularity if α /. An argument similar to that in [4] would need an L version of the boundary recovery result of Brown (see also [5]) of the type γ γ L ( Ω) Cρ. 4. Beltrami equations and fractional Sobolev spaces. This section is devoted to investigate how quasiconformal mappings interplay with fractional Sobolev spaces. We face three different goals. First, given a Beltrami coefficient µ W α, 0 (C), we find β (0, α) such that for any K-quasiconformal mapping φ the composition µ φ, which is another Beltrami coefficient with the same ellipticity bound, belongs to W β, (C). Secondly, we obtain the optimal (at least when α ) Sobolev regularity for the homeomorphic solutions to the equation f = µ f + ν f under the assumptions of ellipticity and Sobolev regularity for the coefficients. Finally, we obtain bounds for the complex geometric optics solutions. Many properties of planar quasiconformal mappings rely on two integral operators, the Cauchy transform, (30) Cϕ(z) = π and the Beurling transform, (3) Tϕ(z) = π lim ε 0 ϕ(w) (w z) da(w). w z ε ϕ(w) (w z) da(w). Their basic mapping properties are well known and can be found in any reference concerning planar quasiconformal mappings, see for instance [3, 9, ]. For s (, ) we will denote by T L s (C) the norm of T as a bounded operator in L s (C). We recall also their relation with complex derivatives which holds for any ϕ C 0 (C). Cϕ = ϕ, T( ϕ) = ϕ

16 64 Albert Clop, Daniel Faraco and Alberto Ruiz 4.. Composition with quasiconformal mappings. Let µ be a compactly supported Beltrami coefficient, satisfying µ K K + χ D = κ χ D. Further, assume that µ W α, (C) and µ W α, (C) Γ 0 for some α > 0 and some Γ 0 > 0. Let φ : C C be a planar K-quasiconformal mapping. In this section, we look for those β > 0 such that µ φ W β, (C). We need to recall a local version of a lemma due to Fefferman and Stein, see [36] and [5, Proposition.4]. The proof follows from Vitali covering Lemma, exactly as in [36]. By Mf we denote the Hardy-Littlewood maximal function, Mf(x) = sup D D f(z) da(z), where the supremum runs over all disks D with x D, while M Ω f denote its local version, that is, M Ω f(x) = sup f(z) da(z), D D where the supremum is taken over all discs D with x D Ω. Lemma 4.. Let < p < and ω 0 a locally integrable function. Then M Ω f(x) p ω(x)da(x) f(x) p Mω(x)dA(x). Ω We can now prove the main result of this section. Proposition. Let K. Let µ W α, (C) for some α (0, ), and assume that µ K K+ χ D. Let φ : C C be any K-quasiconformal mapping, conformal out of a compact set, and normalized so that φ(z) z 0 as z. Then whenever β < α K. Moreover, Ω µ φ W β, (C) µ φ W β, (C) C µ K W α, (C), for some constant C > 0 depending only on α, β and K. Proof. It is clear that µ φ belongs to L (C), so since W α, agrees with the Besov space Bα,, it suffices to show the convergence of the integral µ(φ(z + w)) µ(φ(z)) da(z)da(w) C C w +β for every β < α K. First of all, for large w there is nothing to say since µ(φ(z + w)) µ(φ(z)) da(z)da(w) w > C = w > w > w +β w +β C 4 µ L (C) φ (D) w +β µ(φ(z + w)) µ(φ(z)) da(z)da(w) da(w) = 4π µ L (C) φ (D). β

17 Stability of Calderón s Problem 65 Then we are left to bound the integral µ(φ(z + w)) µ(φ(z)) da(z)da(w). w C w +β As µ has support in D, and w, the difference µ(φ(z+w)) µ(φ(z)) is supported in the -neighbourhood of φ (D), that is, F = {z C : d(z, φ (D)) }. Indeed, φ(z) D if and only if z φ (D) F, while φ(z + w) D if and only if z φ (D) w. But if z = φ (ζ) w for some ζ <, d(z, φ (D)) = inf ξ < z φ (ξ) = inf ξ < φ (ζ) w φ (ξ) φ (ξ) w φ (ξ) = w, so also φ (D) w F. In other words, if z / F then µ(φ(z)) = µ(φ(z + w)) = 0, and we are reduced to bound µ(φ(z + w)) µ(φ(z)) (3) da(z)da(w). w F w +β Note also that, by Koebe s 4 Theorem, we have the inclusions φ(d) 4D and φ (4D) 6D, so that F 7D. To bound (3) the local behavior of µ is important, so we will use condition (7) for the function µ. But before, recall that µ W α, (C) L (C), so that by interpolation we obtain µ W αθ, θ (C) for each θ (0, ), with the estimates µ W αθ, θ (C) C µ θ L µ θ W α, (C). Thus, by (7), for every λ (0, αθ) there exists a function g = g λ L p λ (C), p λ = λ+( α)θ, such that µ(ζ) µ(ξ) ζ ξ λ (g(ζ) + g(ξ)) at almost every ζ, ξ C. The choice θ = /K and Lemma 3. also gives us L p λ estimates, (33) g λ L p λ(c) C µ W α K,K (C) C µ K L µ K W α, (C). with C > 0. It follows that (34) µ(φ(z + w)) µ(φ(z)) w λ ( ) λ φ(z + w) φ(z) (g(φ(z + w)) + g(φ(z))). w Next we recall that quasiconformal mappings are quasisymmetric (see for instance [9] or [33]). That is, for K there exists an increasing homeomorphism η K : R R such that for any K-quasiconformal mapping φ, and for any a, z, z C, we have φ(z ) φ(a) φ(z ) φ(a) η K ( ) z a. z a

18 66 Albert Clop, Daniel Faraco and Alberto Ruiz Thus (35) ( ) (diamφ(d(a, r))) 4 max φ(z ) φ(a) z a =r ( ) 4η K () min φ(z ) φ(a) z a =r 4 η() φ(d(a, r)) = C K J(z, φ) da(z), π D(a,r) for some C K > depending only on K. We now plug (35) into (34), and use that λ <, ( ) λ ( ) λ φ(z + w) φ(z) diamφ(d(z, w )) C K w diam D(z, w ) ( ) λ C K J(ζ, φ) da(ζ) D(z, w ) D(z, w ) ( ) C K J(ζ, φ) λ da(ζ). D(z, w ) D(z, w ) At the last step we used the reverse Hölder inequality for jacobians of quasiconformal mappings, which holds uniformly in λ because λ (0, ) (for a precise result see [0, Theorem ]). Thus, if Ω = { z C : d(z, φ (D)) } then ( ) λ φ(z + w) φ(z) C K (M Ω J λ (z)) w where M Ω J λ (z) denotes the local Hardy-Littlewood maximal function M Ω at the point z of J(, φ) λ. Note also that Ω 8D by Koebe s Theorem. By symmetry, we could also write M Ω J λ (z + w) instead of M Ω J λ (z), so the integral at (3) is bounded from above by M Ω J λ (z + w)g(φ(z + w)) + M Ω J λ (z)g(φ(z)) (36) C K da(z)da(w), w F w +β λ and this reduces our job to find bounds for C K (37) M Ω J λ (z)g(φ(z)) da(z), λ β Ω whenever λ > β. Indeed, we simply divide the integral of (36) into two terms, one in z (for which the bound (37) is obvious) and one in z + w. For the second one, we note that if z F and w then z +w Ω and after a change of coordinates we obtain M Ω J λ (z + w)g(φ(z + w)) C K da(z)da(w) w F C K = C K λ β w +β λ w Ω Ω M Ω J λ (ζ)g(φ(ζ)) da(ζ) M Ω J λ (ζ)g(φ(ζ)) da(ζ) da(w) w +β λ provided that λ > β, and where C K may have changed, but still depends only on K, as claimed in (37).

19 Stability of Calderón s Problem 67 To finish the proof, we will use Lemma 4. and the fact that jacobians of quasiconformal mappings are A weights. This requires two auxiliary indexes r, s >, chosen as follows: For each λ (β, α K ), we have Kλ + ( α) <, whence there exists numbers s such that (38) < s < Kλ + ( α). For instance, (39) s = + ( ) Kλ + ( α). (40) Further, since β < λ we get < Kλ + ( α) < + α Kβ (α Kβ) therefore, by choosing α Kβ < /, we can assume that s <. Note that s and α Kβ are comparable quantities. Recall that p λ =. Now, the choice (39) guarantees us that K Kλ+( α) p λ Ks = s >. Hence we can find numbers r satisfying (4) < (4) as for instance r + λs(k ) < p λ Ks, r + λs(k ) = 3 s. r Again, the difference +λs(k ) is comparable to α λk. By denoting α Kβ = ǫ > 0, the particular choice gives us the following parameters: s = + and similarly ǫ 4 ǫ r = + (K )β + M ǫ λ = β + α Kβ K + λs(k ) = + (K )β + M ǫ + (K )β for small enough ǫ r r = + (K )β + M ǫ + for small enough ǫ (K )β where M, M are positive constants depending only on K. Once the parameters have been chosen, we can start bounding the integral at (37). Since we can not

20 68 Albert Clop, Daniel Faraco and Alberto Ruiz work in L, we first bring s into the estimates by Hölder s inequality, M Ω J λ (z)(g φ(z)) da(z) = M Ω J λ (z)(g φ(z)) χ Ω (z)da(z) Ω Ω (M Ω J λ (z)) s (g φ(z)) s χ Ω (z)da(z) Ω }{{} I Now Lemma 4. provide us with a constant C (s) to obtain I C (s) J λ (z) s M((g φ) s χ Ω )(z)da(z). Ω s Ω s. Note that C (s) blows up only as s, that is, as λ α K due to (39). Now, by Hölder s inequality with exponent r, one gets ( I J λ (z) s ( M((g φ) s χ Ω )(z) ) ) ( ) r r da(z) J λ (z) s r da(z). Ω Ω The first inequality at (4) guarantees that the weight J λ (z) s = J(z, φ) λs belongs to the Muckenhoupt class A r (see [0] or [9, Theorem 3.4.]), with constant (43) J s λ Ar C(K) r λs(k ) < C(K) ǫ due to (38), (39) and (4). We can use the weighted L r inequality for the maximal function and a change of coordinates to see that J λ (z) s ( M((g φ) s χ Ω )(z) ) r da(z) C J λ (z) s (g φ(z)) sr da(z) C Ω = C J(w, φ ) λs g(w) sr da(w). φ(ω) The precise behavior for C comes from [] (see also [37, Theorem.]), C r = M Lr (Jλ s da) C r r Js λ r A r ( ) C(K) C + (K )β + M ǫ ǫ C(K), βǫ where C(K) is a positive constant that depends only on K. Summarizing, we get for the integral at (37) the bound C /r Ω s ( J(z, φ) λs da(z) Ω ) s sr ( ) rs J(w, φ ) λs g(w) sr da(w) } φ(ω) {{ } II Now, the second inequality at (4) gives us that p λ > rs. Thus Hölder s inequality is justified and we get II ( g(w) p λ da(w) φ(ω) ) p λ ( J(w, φ ) p λ ( λs) φ(ω) p λ rs da(w) ) p λ rs p λ rs..

21 Stability of Calderón s Problem 69 The first integral above is finite since g L p λ. To see the finiteness of the second integral, observe that φ is a K-quasiconformal mapping, hence by Astala s Theorem [8] the p λ( λs) p λ rs -th power of its Jacobian determinant J(, φ ) will be locally integrable provided that this exponent does not exceed K K. But p λ ( λs) p λ rs < K K r < p λ ( + λs(k )) sk which comes again from the second inequality at (4). Furthermore, K K p λ( λs) p λ rs M 3ǫ where M 3 > 0 depends only on K. Thus we have that II C(K)ǫ K g L p λ(φ(ω)) where the constant C(K) depends only on K. This means that (37) has the upper bound C /r Ω ( ) s α K β J(z, φ) λs s sr da(z) C(K)ǫ K g L p λ(φ(ω)) Ω ( Ω C(K) ( Ω rs βǫ3 /K Ω C(K) β ǫ 3 /K g L p λ(φ(ω)) C(K) Ω rs βǫ3 /K Ω Ω ) J(z, φ) λs s sr da(z) g L pλ(φ(ω)) ) λ( r) J(z, φ)da(z) g L p λ(φ(ω)) where we have used that λs <, the area distortion theorem of Astala [8] and the fact that Ω C(K). Using (33), one finally obtains for the square root of the integral at (3) the bound Since µ /K L (C) as desired. C(K) β / ǫ µ /K (3 /K) L (C) µ /K W α, (C). <, the obtained inequality for the nonhomogeneous norms is µ φ W β, (C) C(K) β(α Kβ) 3 /K µ K W α, (C), Remark 3. The condition β < α K is by no means sharp. This is clear when α is close to. As promised in the introduction this will be a matter of a forthcoming work. 4.. Regularity of homeomorphic solutions. We start by recalling the basic result on the existence of homeomorphic solutions to Beltrami type equations. In absence of extra regularity the integrability of the solutions comes from the work of Astala [8]. We recall the proof in terms of Neumann series since it will be used both in this section and in the sequel.

22 70 Albert Clop, Daniel Faraco and Alberto Ruiz Lemma 4.. Let µ, ν be bounded functions, compactly supported in D, such that µ(z) + ν(z) K K+ at almost every z C. The equation (44) f = µ f + ν f admits only one homeomorphic solution φ : C C, such that φ(z) z = O(/ z ) as z. Further, if p ( K K+, K K ) then the quantity φ L p (C) + φ L p (C) is bounded by a constant C = C(K, p) that depends only on K and p. Proof. Put φ(z) = z + Ch(z), where h is defined by (I µ T ν T)h = µ + ν. and C and T denote, respectively, Cauchy and Beurling transforms. Since T is an isometry in L (C), one can construct such a function h as Neumann series h = (µt + νt) n (µ + ν) n=0 which obviously defines an L (C) function. By Riesz-Thorin interpolation theorem, lim T L p p (C) =, it then follows that h L p (C) for every p > such that T Lp (C) < K+ K. Hence, the Cauchy transform Ch is Hölder continuous (with exponent p ). Further, since h is compactly supported, we get φ(z) z = Ch(z) C z, and in fact φ z belongs to W,p (C) for such values of p. A usual topological argument (see for instance [9, Chapter 5]) proves that φ is a homeomorphism. For the uniqueness, note that if we are given two solutions φ, φ as in the statement then (φ φ ) = 0 so that φ φ (z) z is holomorphic on C and vanishes at infinity. Now we recall a remarkable result from [0], which says that I µ T ν T : L p (C) L p (C) defines a bounded invertible operator whenever p ( K K+, K K ). Further, for the norm of the inverse operator we have the following estimate, Thus, if p ( K K+, K K ) Therefore (I µ T ν T) Lp (C) C(K, p). h Lp (C) C(K, p) µ + ν Lp (C) C(K, p). φ L p (C) + φ L p (C) = Th L p (C) + h L p (C) C(K, p) since T is a bounded operator in L p (C). Once we know about the existence of homeomorphic solutions, it is time to check their regularity when the coefficients belong to some fractional Sobolev space. Theorem 4.3. Let α (0, ), and suppose that µ, ν W α, (C) are Beltrami coefficients, compactly supported in D, such that µ(z) + ν(z) K K +. at almost every z D. Let φ : C C be the only homeomorphism satisfying φ = µ φ + ν φ

23 Stability of Calderón s Problem 7 and φ(z) z = O(/z) as z. Then, φ(z) z belongs to W +θα, (C) for every θ (0, K ), and ( ) D +θα (φ z) L (C) C µ θ W α, (C) + ν θ W α, (C) for some constant C = C(K, θ, α). Proof. We consider a C function ψ, compactly supported inside of D, such that 0 ψ and ψ =. For n =,,... let ψ n (z) = n ψ(nz). Put µ n (z) = µ(w)ψ n (z w)da(w), and ν n (z) = C C ν(w)ψ n (z w)da(w). It is clear that both µ n, ν n are compactly supported in n+ n D, µ n(z) + ν n (z) K K+, µ n µ W α, (C) 0 and ν n ν W α, (C) 0 as n. Indeed there is convergence in L p for all p (, ). Thus, by interpolation we then get that for any 0 < θ < lim µ n µ n W αθ, θ + ν n ν (C) W αθ, θ = 0 (C) and in particular, the sequences D αθ µ n and D αθ ν n are bounded in L θ (C). Let φ n be the only K-quasiconformal mapping φ n : C C satisfying (45) φ n = µ n φ n + ν n φ n and normalized by φ n (z) z = O n (/z) as z. By the construction in Lemma 4., φ n (z) = z + Ch n (z) where h n is the only L (C) solution to h n = µ n Th n + ν n Th n + (µ n + ν n ), and Ch n denotes the Cauchy transform. As in Lemma 4., h n belongs to L p (C) for all p ( K K+, K K ) and (46) h n L p (C) C(K, p) with a constant C(K, p) that depends on K and the product ( K K K p)(p K+ ). In particular, φ n z is a bounded sequence in W,p (C). Let us denote H n (z) = Ch n (z) = φ n (z) z. We now write equation (45) as H n = µ n H n + ν n H n + µ n + ν n and take fractional derivatives. If β = αθ, we can use Lemma 3. (a) to find two functions E β, F β such that D β H n µ n D β H n ν n D β H n = D β µ n H n + E β + D β ν n H n + F β. Now recall that we have D β ϕ = D β ϕ and similarly for. Further, if ϕ is real then D β ϕ is also real. Thus D β H n µ n D β H n ν n D β H n = D β µ n H n + E β + D β ν n H n + F β.

24 7 Albert Clop, Daniel Faraco and Alberto Ruiz Equivalently, since T =, (47) (I µ n T ν n T) ( D β H n ) = D β µ n H n + D β ν n H n + E β + F β. For E β and F β we have precise L estimates. To see this, choose p = θ, and then let p be such that < p < K K and p + p =. Observe that this forces 0 < θ < K, and further (48) K K p = ( ) (K )( θ) ( Kθ) C K K θ. Now, by Lemma 3. there exists C 0 = C 0 (β, p, p ) such that (49) E β L (C) C 0 D β µ L p (C) H n L p (C), and similarly (50) F β L (C) C 0 D β ν L p (C) H n L p (C). This says us that the right term at (47) is in fact an L (C) function, whose L (C) norm is bounded from above by (C 0 + ) ( D β µ n L p (C) + D β ν n L p (C) ) Hn L p (C). Now, recall that the operator I µ n T ν n T is continuously invertible in L (C), and a Neumann series argument shows that the norm of its inverse is bounded by (K + ). Thus, D β H n L (C) (C 0 + ) K + (C 0 + ) K + ) ( D β µ n + D β ν L θ n H (C) L θ n (C) L p (C) ( ) µ n θ W α, (C) + ν n θ W α, (C) H n L p (C) where C 0 = C 0 (β, p, p ) is the constant in (49). As n, we have the uniform bound (46), H n L p (C) = Th n L p (C) C p h n L p (C) C where now the constant C = C (K, θ) depends on K and K θ. Thus, we obtain for D β H n L (C) the upper bound ( ) (C 0 + )C (K, θ) µ θ W α, (C) + ν θ W α, (C). By passing to a subsequence we see that D β H n converges in W, (C), and as a consequence φ z belongs to W +β, (C). Further, we have the bounds ( ) D +θα (φ z) L (C) C µ θ W α, (C) + ν θ W α, (C) with C depending only on K, α and α K β.

25 Stability of Calderón s Problem Regularity of complex geometric optics solutions. We are now ready to give precise bounds on the Sobolev regularity of the complex geometric optics solutions to the equation f = µ f introduced in Theorem.. For this, the following lemma will be needed. Lemma 4.4. Let Ω C be any domain. If f W α,p (Ω) and ϕ C (C) is bounded, then the multiplier f ϕf is bounded from W α,p (Ω) to itself, and ( ϕf W α,p (Ω) ϕ L (Ω) + Dϕ ) α L (Ω) f W α,p(ω), ϕ L (Ω) whenever 0 α, < p <. The proof follows easily by interpolation. Theorem 4.5. Let µ W α, (C) be such that µ K K+ χ D and µ W α, (C) Γ 0. Let f = f µ (z, k) the complex geometric optics solutions to the equation f = µ f. For any 0 < θ < K we have that f W +θα, loc (C). Further, we have the estimate D +αθ (f µ )(, k) L (D) e C(K) k ( + Γ θ ) 0 whenever 0 < θ < K. Proof. The case k = 0 is trivial. For k 0, the existence and uniqueness of the complex geometric optics solutions comes from [, Theorem 4.] (see Theorem. in the present paper). It is shown in [, Lemma 7.] that f may be represented as f(z, k) = e ikφ(z,k) where φ : C C is the only W, loc (C) homeomorphism solving (5) φ(z) = µ(z) k k e k(φ(z, k)) φ(z) and such that (5) φ(z) z C (K), z. z Let us recall that e k (w) = e ikw ikw is a unimodular function, whence e k (φ(z)) =. We will first deduce the smoothness of φ from that of its Beltrami coefficient µ e k (φ), see (5), with the help of Theorem 4.3. For the ellipticity there is nothing to say since µ(z)e k (φ(z, k)) = µ(z) K K +. For the Sobolev regularity, we will use that e k (φ(z, k)) = e k (φ(z, k) z)e k (z),

26 74 Albert Clop, Daniel Faraco and Alberto Ruiz which is more convenient since for φ(z, k) z we have global estimates. We then describe the Beltrami coefficient of (5) as ( ) µ e k (φ) = µ e k (φ z) e k (z) + µ e k (z). Using Lemma 4.4 with ϕ(z) = e k (z), µ e k (φ) W α, (C) µ (e k (φ z) )e k W α, (C) + µ e k W α, (C) ( + k ) α µ g W α, (C) + ( + k ) α µ W α, (C) where g = e k (φ z). For the first term above we use the fractional Leibnitz rule (Lemma 3.) to get that D α (µg) L (C) D α g L (C) µ L (C) + (C 0 + ) D α µ L (C) g L (C). The bound for D α g L (C), will be found by interpolation. Bounds for D g come easily from Lemma 4.. Indeed, φ(z, k) z belongs to W,p (C) for every K K+ < p < K K, and in fact, by using the chain rule whence g(z) k (φ(z, k) z) ( + g(z) ) 3 k D(φ(z, k) z), D g L p (C) k C(K, p) for every K K+ < p < K K. The Lp bounds for g (now with p > ) follow from the decay estimate (5) and the global boundedness of g. We obtain that g L p (C) for any p >, with g L p (C) ( + k ) C(K, p). If we now let max{ K K+, α} < p < and = α p + α p, then p >. Therefore g L p (C), D g L p (C) and D α g L (C) D g α L p (C) g α L p (C) C(K)( + k ) α. Summarizing, D α (µg) L (C) C Γ 0 + C(K)( + k ) α. By Theorem 4.3, the smoothness of φ(, k) can be recovered by that of its coefficient, and we get (53) D +αθ (φ(z, k) z) L (C) C(K, θ, α)(γ 0 + ( + k ) α ) θ, for any 0 < θ < K. Now the job is to use the above estimates to get local bounds for the derivative D(e ikφ ) in the W αθ, norm. For this, we write again (54) e ikφ = e ik(φ z) e ikz By letting h(z, k) = e ik(φ(z,k) z), by the chain rule one gets, for the first order derivatives ) (e ikφ ) = ik e (h ikz (φ z) + h z. But and h have obvious local Sobolev bounds, and furthermore e ikz is C. Thus by Lemma 4.4 we get for any disk D, for some C = C(D), that (55) (e ikφ ) W αθ, (D) k ( + k ) αθ e C k h (φ z) + h z W αθ, (D)