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.

7 pages

References in corpus (3)

• Beltrami equation with coefficient in Sobolev and Besov spaces
• A T(P) theorem for Sobolev spaces on domains
• Sobolev regularity of the Beurling transform on planar domains