A removability theorem for Sobolev functions and detour sets
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Mathematische Zeitschrift
A removability theorem for Sobolev functions and detour sets Dimitrios Ntalampekos1 Received: 16 September 2017 / Accepted: 12 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We study the removability of compact sets for continuous Sobolev functions. In particular, we focus on sets with infinitely many complementary components, called “detour sets”, which resemble the Sierpi´nski gasket. The main theorem is that if K ⊂ Rn is a detour set and its complementary components are sufficiently regular, then K is W 1, p -removable for p > n. Several examples and constructions of sets where the theorem applies are given, including the Sierpi´nski gasket, Apollonian gaskets, and Julia sets. Keywords Removability · Sobolev functions · Hölder domains · Detour sets · Sierpi´nski gasket Mathematics Subject Classification Primary 46E35; Secondary 30C65
1 Introduction 1.1 Background In this paper we study the removability of sets for Sobolev functions in Rn . The problem originally arises from the problem of removabilty of sets for (quasi)conformal maps. Definition 1.1 A compact set K ⊂ U ⊂ R2 is (quasi)conformally removable inside the domain U if any homeomorphism of U , which is (quasi)conformal on U \K , is (quasi)conformal on U . (Quasi)conformal removability of sets is of particular interest in Complex Dynamics, since it provides a tool for upgrading a topological conjugacy between two dynamical systems to a (quasi)conformal conjugacy. Another application of removability results is in the problem conformal welding. The relevant result is that if a Jordan curve ∂Ω is (quasi)conformally removable, then the welding map that arises from ∂Ω is unique, up to Möbius transformations.
The author was partially supported by NSF grant DMS-1506099.
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Dimitrios Ntalampekos [email protected] Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY 11794-3651, USA
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D. Ntalampekos
However, the converse is not known, in case ∂Ω has zero area. We direct the reader to [27], and the references therein, for a comprehensive survey on the topic. A stronger notion of removability is the notion of Sobolev W 1,2 -removability. Recall that for an open set U ⊂ Rn and p ≥ 1 we say that a function f lies in the Sobolev space W 1, p (U ) if f ∈ L p (U ) and there exist functions ∂i f ∈ L p (U ), i = 1, . . . , n, such that for every smooth function φ : Rn → R with compact support in U , we have ˆ ˆ ∂i f · φ = − f · ∂i φ (1.1) U
U ∂φ ∂ xi
denote the partial derivatives of φ in the coordinate for all i = 1, . . . , n, where ∂i φ = directions of Rn . We give a preliminary definition of W 1, p -removability in Rn : We say that a compact set K ⊂ U ⊂ Rn is W 1, p -removable inside a domain U , if any function that is continuous in U and lies in W 1, p (U \K ), also lies in W 1, p (U ). In other words, W 1, p (U \K ) ∩ C 0 (U ) = W 1, p (U ) ∩ C 0 (U ) as sets. Here, C 0 (U ) denotes the space of continuous functions on U . It is immediate to chec
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