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

TL;DR

This paper establishes a precise equivalence between the regularity of the Beurling transform of a domain's characteristic function and the boundary's outward normal in certain Sobolev and Besov spaces, linking boundary smoothness to transform behavior.

Contribution

It proves a new if-and-only-if condition connecting the Beurling transform's Sobolev regularity with the boundary normal's Besov regularity for planar domains.

Findings

Beurling transform in W^{a,p} implies boundary normal in B_{p,p}^{a-1/p}

Boundary normal in B_{p,p}^{a-1/p} implies Beurling transform in W^{a,p}

Boundedness of Beurling transform in W^{a,p} is characterized by boundary normal regularity

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<a\leq 1, 1<p<\infty such that ap>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.