  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.
