Weighted Sobolev estimates of the truncated Beurling operator

Given a bounded planar domain $D$ with $W^{k+1,\mathrm{\infty }}$ boundary, $k\in {\mathbb{Z}}^{+}$, and a weight $\mu \in A_p,1<p<\mathrm{\infty }$, 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.