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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 thecorresponding truncated Beurling transform is a bounded operator sending $W^{k,p}(D, \mu)$ into itself. Weighted Sobolev estimates for other Cauchy-typeintegrals are also obtained.

Weighted Sobolev estimates of the truncated Beurling operator