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Analytic Capacity and Quasiconformal Mappings with W1,2 Beltrami

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Analytic Capacity and Quasiconformal Mappings with W1,2 Beltrami Analytic capacity and quasiconformal mappings with W 1,2 Beltrami coefficient Albert Clop Xavier Tolsa ∗ Abstract We show that if φ is a quasiconformal mapping with compactly sup- ported Beltrami coefficient in the Sobolev space W 1,2, then φ preserves sets with vanishing analytic capacity. It then follows that a compact set E is removable for bounded analytic functions if and only if it is removable for bounded quasiregular mappings with compactly supported Beltrami coefficient in W 1,2. 1 Introduction A Beltrami coefficient is a measurable function µ such that kµk∞ < 1. Given an open set Ω ⊂ C, we say that f :Ω → C is µ-quasiregular if it belongs to the 1,2 Sobolev space Wloc (Ω) and satisfies the Beltrami equation ∂f(z) = µ(z) ∂f(z), a.e.z ∈ Ω. If moreover f is a homeomorphism, then we call it µ-quasiconformal. For any K ≥ 1, we say that f is K-quasiregular (or K-quasiconformal if f is homeo- K−1 morphism) for some Beltrami coefficient µ satisfying kµk∞ ≤ K+1 . Several works have focussed in the question of how these mappings distort mea- sures and capacities. For instance, Ahlfors (see [Ah1]) proved that they always preserve sets of zero area. In a remarkable paper, Astala [As] obtained deep estimates for the area distortion under K-quasiconformal mappings. More pre- cisely, if φ is any (conveniently normalized) K-quasiconformal mapping, then ∗A. C. was supported by grants MTM2007-60062 and 2005-SGR-00774; and X. T. by MTM2007-62817 and and 2005-SGR-00774. AMS (2000) Classification. Primary 30C60, 35J15, 35J70. Keywords: quasiconformal, Hausdorff measure, removability, analytic capacity 1 one has the estimate 1 |φ(E)| ≤ C |E| K where the constant C depends only on K. As a consequence, the author ob- tained also sharp results on integrability of K-quasiconformal mappings, which in turn led to the bounds on K-quasiconformal distortion of Hausdorff dimen- sion. Namely, for any K-quasiconformal mapping φ and any compact set E, µ ¶ µ ¶ 1 1 1 1 1 1 1 − ≤ − ≤ K − . (1) K dim(E) 2 dim(φ(E)) 2 dim(E) 2 Moreover, in [As] the author shows the sharpness of both inequalities. It is well known that sometimes the regularity of the Beltrami coefficient µ is inherited by the mapping itself. For instance, when µ is a compactly sup- ported C∞ function, then every µ-quasiconformal mapping φ is also C∞. As a consequence, φ is locally bilipschitz, and then some set functions like Hausdorff measures, Riesz and Bessel capacities, are preserved. Nevertheless, there are other situations which, even far from µ ∈ C∞, give interesting improvements of equation (1). For instance, when µ belongs to the class VMO of functions of vanishing mean oscillation, then dim(φ(E)) = dim(E). That is, the corresponding µ-quasiconformal mappings φ do not distort Haus- dorff dimension (see for instance [Iw]). However, nothing is known on the ratio between the Hausdorff measures Ht(E) and Ht(φ(E)), for any t ∈ [0, 2]. In this context, of special interest is the assumption that µ is a compactly sup- ported function in the Sobolev class W 1,2 (notice that this implies µ ∈ VMO). For such Beltrami coefficients, it is shown in [CFMOZ] that the corresponding µ-quasiconformal mappings φ preserve sets with zero length, that is H1(E) = 0 ⇐⇒ H1(φ(E)) = 0, (2) The proof of this fact uses some BMO removability techniques, related to both the Cauchy-Riemann (i.e. ∂) and the Beltrami (∂ − µ ∂) differential operators. The main tool is an extended version of Weyl’s lemma. Recall that classi- cal Weyl’s Lemma asserts that distributional solutions to the Cauchy-Riemann equation are actually analytic functions. In the more general case of the Bel- trami equation [CFMOZ, Theorem 1], an analogous result can be given provided that the Beltrami coefficient belongs to W 1,2. 2 Theorem. Let µ be a compactly supported Beltrami coefficient in the Sobolev 1,2 p space W (C). Let f ∈ Lloc(C) for some p > 2, and suppose that h∂f − µ ∂f, ϕi = 0 whenever ϕ ∈ C∞ is compactly supported. Then, f is µ-quasiregular. In [CFMOZ], similar arguments to those in (2), replacing BMO by VMO, allowed the authors to prove that if µ ∈ W 1,2 is any compactly supported Beltrami coefficient, and φ is µ-quasiconformal, then H1(E) is σ-finite ⇐⇒ H1(φ(E)) is σ-finite. (3) Furthermore, these mappings φ are shown to map 1-rectifiable sets to 1-rectifiable sets (and purely 1-unrectifiable sets to purely 1-unrectifiable sets). As we shall see in this paper, all these facts have interesting consequences when studying removability problems for bounded µ-quasiregular mappings, that is, the µ-quasiregular counterpart for the problem of Painlev´e.Recall that a com- pact set E is said to be removable (for bounded analytic functions) if for any open set Ω ⊃ E, every bounded function f :Ω → C, analytic on Ω \ E, admits an analytic extension to the whole of Ω. The problem of Painlev´e consists of giving metric and geometric characterizations of these sets. When studying removable sets, it is natural to talk about analytic capacity. Recall that given a compact set E, the analytic capacity of E is defined as 0 ∞ γ(E) = sup {|f (∞)|; f ∈ H (C \ E), kfk∞ ≤ 1} . Here, by H∞(Ω) we mean the space of bounded analytic functions on the open set Ω, and f 0(∞) = lim z (f(∞) − f(z)). For a set A ⊂ C which may be non z→∞ compact, one defines γ(A) = sup γ(E). E⊂A compact Ahlfors [Ah2] proved that E is removable for bounded analytic functions if and only if γ(E) = 0. Furthermore, it is not difficult to show that γ(E) ≤ C H1(E), while dim(E) > 1 implies γ(E) > 0. It took long time to have a precise geometric characterization of the zero sets for γ. In [Da1], G. David proved that if E has finite length then γ(E) = 0 ⇐⇒ E is purely 1-unrectifiable. Later, in [To2], X. Tolsa characterized sets with vanishing analytic capacity in terms of Menger curvature (see Theorem 5 below for more details). 3 In this paper, as well as in [CFMOZ], our main objects of study are the remov- able singularities for bounded solutions to a fixed Beltrami equation. Namely, we say that a compact set E is removable for bounded µ-quasiregular mappings, or simply µ-removable, if for any open set Ω, any bounded function f :Ω → C, µ-quasiregular on Ω \ E, admits a µ-quasiregular extension to the whole of Ω. By means of Stoilow’s factorization Theorem, one easily shows that E is µ- removable if and only if γ(φ(E)) = 0 for any µ-quasiconformal mapping φ. In connection with this question, the following result is proved in [CFMOZ]. Theorem. Let µ ∈ W 1,2(C) be a compactly supported Beltrami coefficient, and let φ : C → C be a µ-quasiconformal mapping. Then, γ(E) = 0 ⇐⇒ γ(φ(E)) = 0 (4) for any compact set E with σ-finite H1(E). Pekka Koskela suggested us that the σ-finiteness assumption might be removed in the preceding result. In this paper we do the job. Theorem 1. Let µ ∈ W 1,2(C) be a compactly supported Beltrami coefficient, and let φ : C → C be a µ-quasiconformal mapping. Then, γ(E) = 0 ⇐⇒ γ(φ(E)) = 0 for any compact set E. It follows from Theorem 1 that if µ ∈ W 1,2 is compactly supported, then being removable and being µ-removable are equivalent notions. Corollary 2. Let µ ∈ W 1,2 be a compactly supported Beltrami coefficient. Then, a compact set E is removable for bounded µ-quasiregular mappings if and only if γ(E) = 0. Theorem 1 implies that, given a compactly supported Beltrami coefficient µ ∈ W 1,2(C), the corresponding µ-quasiconformal mappings preserve the removable sets for bounded analytic functions. This fact is closely related to a question of J. Verdera [Ve1] on the preservation of removable sets under some planar homeomorphisms. More precisely, the author wondered how analytic capacity is distorted under bilipschitz mappings. Recall that a mapping φ : C → C is said to be L-bilipschitz if 1 |z − w| ≤ |φ(z) − φ(w)| ≤ L|z − w| L for any pair of points z, w ∈ C. This question was solved in [To2]: 4 Theorem. Let φ : C → C be an L-bilipschitz mapping. Then, γ(φ(E)) ' γ(E) (5) with constants that depend only on L. Furthermore, it is shown in [To2] that any planar homeomorphism φ : C → C satisfying (5) must be a bilipschitz mapping. It is well known that L-bilipschitz mappings are µ-quasiconformal for some Beltrami coefficient µ with kµk∞ de- pending only on L, but in general this does not imply any W 1,2 regularity for µ. In fact, there is not a precise description of the class of compactly sup- ported Beltrami coefficients µ that produce bilipschitz µ-quasiconformal map- pings. It was remarked in [CFMOZ, Example 4] that there are non bilipschitz µ-quasiconformal mappings with compactly supported µ ∈ W 1,2. At the same 1 time, the example µ(z) = 2 χD(z) gives a bilipschitz µ-quasiconformal mapping with µ∈ / W 1,2. Thus, there is no relation between bilipschitz µ-quasiconformal mappings and Beltrami coefficients µ ∈ W 1,2. For the proof of the Theorem 1, our main tool is the following improved version of the preceding theorem on the bilipschitz invariance of analytic capacity.
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