Regularity of C¹ and Lipschitz domains in terms of the Beurling transform

2012/01/25 by Xavier Tolsa, Tolsa, Xavier
Mathematics · #30C62 #35R35 #46E35 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1201.5403

openalex publication_date 2012/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let D be a bounded planar C¹ 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 01, 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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