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arXiv · 1201.5403

Regularity of C^1 and Lipschitz domains in terms of the Beurling transform

Xavier Tolsa
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Abstract

Let D be a bounded planar C^1 domain, or a Lipschitz domain "flat enough", and consider the Beurling transform of 1_D, the characteristic function of D. Using a priori estimates, in this paper we solve the following free boundary problem: if the Beurling transform of 1_D belongs to the Sobolev space W^{a,p}(D) for 0 1, then the outward unit normal N on bD, the boundary of D, is in the Besov space B_{p,p}^{a-1/p}(bD). The converse statement, proved previously by Cruz and Tolsa, also holds. So we have that B(1_D) is in W^{a,p}(D) if and only if N is in B_{p,p}^{a-1/p}(bD). Together with recent results by Cruz, Mateu and Orobitg, from the preceding equivalence one infers that the Beurling transform is bounded in W^{a,p}(D) if and only if the outward unit normal N belongs to B_{p,p}^{a-1/p}(bD), assuming that ap>2.

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Xavier Tolsa. 2012-01-25. Regularity of C^1 and Lipschitz domains in terms of the Beurling transform. https://arxiv.org/abs/1201.5403

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