Wavelet resolution and Sobolev regularity of Calder\'on-Zygmund operators on domains

Given a uniform domain $\Omega \subset {\mathbb R}^d$, we resolve each element of a suitably defined class of Calder\`on-Zygmund (CZ) singular integrals on $\Omega$ as the linear combination of Triebel wavelet operators and paraproduct terms. Our resolution formula entails a testing type characterization, loosely in the vein of the David-Journ\'e theorem, of weighted Sobolev space bounds in terms of Triebel-Lizorkin and tree Carleson measure norms of the paraproduct symbols, which is new already in the case $\Omega={\mathbb R}^d$ with Lebesgue measure. Our characterization covers the case of compressions to $\Omega$ of global CZ operators, extending and sharpening past results of Prats and Tolsa for the convolution case. The weighted estimates we obtain, particularized to the Beurling operator on a Lipschitz domain with normal to the boundary in the corresponding sharp Besov class, may be used to deduce quantitative estimates for quasiregular mappings with dilatation in the Sobolev space $W^{1,p}(\Omega)$, $p>2$.