Weighted Sobolev estimates of the truncated Beurling operator

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.