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Removable Singularities for Planar Quasiregular Mappings

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Removable Singularities for Planar Quasiregular Mappings Removable singularities for planar quasiregular mappings Albert Clop PhD Thesis Memoria` presentada per aspirar al grau de doctor en Matematiques` iii iv CERTIFIQUEM que aquesta memoria` ha estat realitzada per Albert Clop, sota la direccio´ dels Drs. Joan Mateu Bennassar i Joan Orobitg Huguet. Bellaterra, octubre del 2006. Joan Mateu Bennassar Joan Orobitg Huguet v Introduction A compact planar set E is said to be removable for bounded analytic functions if for any open set W ⊃ E, any bounded function f : W ! C holomorphic on W n E admits a holomorphic extension to W. It is enough to care about the case W = C. The Painleve´ problem consists on characterizing these sets in metric and geometric terms. The analytic capacity of a compact set E, introduced by Ahlfors [3], is defined by 0 g(E) = sup j f (¥)j; f 2 H(C n E), k f k¥ ≤ 1, f (¥) = 0 . This set function vanishes only on removable sets. Some excellent introductions to g are given by H. Pajot [43] and J. Verdera [55]. As a matter of fact, H1(E) = 0 implies g(E) = 0, while if dim(E) > 1 then g(E) > 0. Thus, 1 is the critical dimension for the Painleve´ problem. Some easier variants of this problem are obtained when replacing L¥ by other classes < of functions, such as BMO or Lipa for 0 a ≤ 1. To specify: • Kaufman [31] showed that E is removable for BMO analytic functions if and only if M1(E) = 0. • Dolzenkoˇ [16] proved that E is removable for Lipa analytic functions if and only if M1+a(E) = 0. • It follows from Verdera's work [54] that E is removable for VMO analytic func- 1 tions if and only if M∗(E) = 0. 1 As it can be seen, the main difference between these variants and the classical Painleve´ problem is that now removable sets are characterized by a Hausdorff content or, in some cases, other set functions, that in any case are not so complicated as analytic ca- pacity. The main purpose of this thesis is the study of removable sets for quasiregular map- pings of the plane. Given a domain W ⊂ C, a function f : W ! C is said to be m- W 1,2 W quasiregular on if it belongs to the Sobolev space Wloc ( ) and its derivatives satisfy at almost every point the so called Beltrami equation, ¶ f = m ¶ f K−1 < for some measurable function m, the Beltrami coefficient, such that kmk¥ = K+1 1. If we forget m and just care about K, then the above PDE is equivalent to the distortion inequality, jD f (z)j2 ≤ K J(z, f ) a.e. z 2 W, where jD f (z)j is the operator norm of the differential matrix of f at the point z, and J(z, f ) is its Jacobian determinant. Then we talk about K-quasiregular mappings. Bijec- tive quasiregular mappings are called quasiconformal. When K = 1 (i.e. if m = 0) we recover, respectively, the classes of analytic functions and conformal mappings on W. For several reasons, this is considered one of the most convenient ways of extending the notion of analycity. As a matter of fact, notice that the distortion inequality has an analog in Rn. However, our interest comes not only from this point of view, but also from the fact that these functions are (at least in the plane) solution to a more or less gen- eral model of an elliptic PDE of first order. As introductory references to this subject, we cite Ahlfors [2], Lehto and Virtanen [37], Reshetnyak [48] and Rickman [49]. In this dissertation we will consider Painleve's´ problem, as well as some of its vari- ants, in the quasiregular context. More precisely, a compact set E is said to be removable for bounded K-quasiregular mappings if for every open set W ⊃ E, any bounded function f : W ! C, K-quasiregular on W n E, admits a K-quasiregular extension to W. It also 2 makes sense to replace the boundedness by other properties, such as being in BMO or in Lipa. Moreover, we can even talk about removable sets for m-quasiregular mappings, when our interest focusses on the Beltrami coefficient, rather than on K. In any case, as in the holomorphic setting, we just need to understand in all situations the case W = C. It follows from Uy's work [53] that if E has positive Lebesgue measure, then E is not removable, neither for K = 1. Conversely, given a compact set E of zero area, and a function f quasiregular on C n E, in order to say that f can be quasiregularly extended to the whole plane, one just must make sure that f extends as a W 1,2 function, in a neighbourhood of E. To explain it, such an f already satisfies a Beltrami equation (or a distortion inequality) almost everywhere. Hence, only the integrability of its distri- butional derivatives has to be checked. In other words, we are facing a simplificated version of a Sobolev removability problem. Chapter 2 is devoted to bounded and BMO K-quasiregular removability. In general, the Stoilow factorization Theorem (see for instance [37]) ensures that if f is K-quasiregular on C n E, then f = h ◦ f, where f : C ! C is K-quasiconformal and h is holomorphic on C n f(E). Moreover, both f and f have the same Beltrami coefficient. Notice also that E Á(E) Á f h Figure 1 f is bounded if and only if h is, and Reimann [46] showed that a similar thing happens with BMO. Therefore, extending f K-quasiregularly is equivalent to extendig h holo- morphically. Thus, in the bounded case we need to check under which conditions is it true that g(f(E)) = 0 3 whenever f is K-quasiconformal. At the same time, for the BMO problem the condition H1(f(E)) = 0 must be checked. In any case, is precisely at this point when the distor- tion problem appears. For instance, one is first led to the dimension distortion formula of K. Astala [4], which says that if dim(E) = t, then 2K dim(E) dim(f(E)) ≤ t0 = 2 + (K − 1) dim(E) for any K-quasiconformal mapping f. Using this inequality, it is shown in [4] that a 1 < 2 sufficient condition for H (f(E)) = 0 (and therefore for g(f(E)) = 0) is t K+1 , or 0 < > 2 equivalently, t 1. In [4] it is also shown that for any real number t K+1 there exists a set E and a K-quasiconformal mapping f such that dim(E) = t and dim(f(E)) > 2 1. Hence, the number K+1 appears as the critical dimension for these K-quasiregular problems of removability. Afterwards, the natural following step to this dimensional setting consists on looking at the K-quasiconformal distortion of Hausdorff measures. For instance, it is well known (see for instance Ahlfors [2, p.33]) that jEj = 0 ) jf(E)j = 0. In this direction we prove the following Theorem. Theorem 1 ([5]). Let E be a compact set, and f a planar K-quasiconformal mapping. Then, 2 1 H K+1 (E) = 0 ) H (f(E)) = 0. 0 Therefore, there is absolute continuity of f∗Ht with respect to Ht both for t0 = 1 and 0 2 t = 2, that is, t = K+1 and t = 2. For general t 2 (0, 2) we cannot prove the absolute ∗ t0 t > 2 continuity of f H with respect to H . However, for t K+1 we obtain a weaker result concerning Bessel capacities. 2 It follows from the above theorem and the BMO invariance that if H K+1 (E) = 0 then E is removable for BMO K-quasiregular mappings. Unexpectedly, for the L¥ problem 4 we can go further and obtain the following result. Theorem 2 ([5]). 2 Let K > 1, and E be a compact set with s-finite H K+1 (E). Then, E is removable for bounded K-quasiregular mappings. In the proof of this Theorem we use several standard quasiconformal tools, together with the optimal integrability result for K-quasiregular mappings obtained by Astala and Nesi [10]. It also uses the fact that the lower 1-dimensional Hausdorff content behaves precisely as the analytic VMO capacity [54]. Moreover, there is a difficult the- orem appearing in [5] which asserts that if y is any K-quasiconformal mapping, and E is a subset of any rectifiable set, then 2 dim(y(E)) > . K + 1 In other words, these sets may not attain the maximal distortion. Furthermore, we can complete the proof by the results on analytic capacity from David [15] and Tolsa [51]. It can be seen that the construction of nonremovable sets for bounded (or BMO) K- quasiregular mappings is equivalent to some critical examples of extremal K-quasiconformal distortion. In that sense, we found sets E and K-quasiconformal mappings f for which 2 > dim(E) = K+1 and g(f(E)) 0. As a consequence, one obtains the following result. Theorem 3 ([5]). > 2 Let K 1. There exists compact sets E of dimension K+1 , non removable for bounded K- quasiregular mappings. Our construction gives examples of compact sets E of dimension t and K-quasiconformal 0 mappings f such that 0 < Ht (f(E)) < ¥. However, we do not know if it is possible to 0 find E and f for which 0 < Ht(E) < ¥ and 0 < Ht (f(E)) < ¥.
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