Smoothness of the Beurling transform in Lipschitz domains

TL;DR

This paper establishes a precise relationship between the boundary regularity of Lipschitz domains and the smoothness of the Beurling transform of their characteristic functions, showing sharp conditions for Sobolev space membership.

Contribution

It proves that boundary regularity in a specific Besov space implies Sobolev regularity of the Beurling transform, extending understanding of its smoothness in Lipschitz domains.

Findings

Beurling transform of characteristic functions lies in Sobolev spaces under boundary Besov regularity.

The result is sharp, indicating optimal boundary conditions for smoothness.

Boundedness of the Beurling transform in Sobolev spaces is characterized by boundary regularity.

Abstract

Let D be a planar Lipschitz domain and consider the Beurling transform of the characteristic function of D, B(1_D). Let 1<p<\infty and 0<a<1 with ap>1. In this paper we show that if the outward unit normal N on bD, the boundary of D, belongs to the Besov space B_{p,p}^{a-1/p}(bD), then the Beurling transform of 1_D is in the Sobolev space W^{a,p}(D). This result is sharp. Further, together with recent results by Cruz, Mateu and Orobitg, this implies that the Beurling transform is bounded in W^{a,p}(D) if N belongs to B_{p,p}^{a-1/p}(bD), assuming that ap>2.