Weighted Sobolev estimates of the truncated Beurling operator

TL;DR

This paper establishes weighted Sobolev boundedness of the truncated Beurling operator on domains with smooth boundaries, extending to other Cauchy-type integrals, thus advancing understanding of weighted integral operators in complex analysis.

Contribution

It provides the first weighted Sobolev estimates for the truncated Beurling operator on domains with smooth boundaries, including bounds for related Cauchy-type integrals.

Findings

Boundedness of the truncated Beurling operator on weighted Sobolev spaces.

Extension of estimates to other Cauchy-type integral operators.

Applicability to domains with $W^{k+1, \infty}$ boundary.

Abstract

Given a bounded planar domain $D$ with $W^{k+1, \infty}$ boundary, $ k\in \mathbb Z^+$, and a weight $\mu\in A_p, 1<p<\infty$, we show that the corresponding truncated Beurling transform is a bounded operator sending $W^{k, p}(D, \mu)$ into itself. Weighted Sobolev estimates for other Cauchy-type integrals are also obtained.