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Given a H\"older uniform domain $\Omega \subset {\mathbb R}^d$, we resolveeach element of a suitably defined class of Calder\'on-Zygmund (CZ) singularintegrals on $\Omega$ as the linear combination of Triebel wavelet operatorsand paraproduct terms. Our resolution formula entails a testing typecharacterization, loosely in the vein of the David-Journ\'e theorem, ofweighted Sobolev space bounds in terms of Triebel-Lizorkin and tree Carlesonmeasure norms of the paraproduct symbols, which is new already in the case$\Omega={\mathbb R}^d$ with Lebesgue measure. Our characterization covers thecase of compressions to $\Omega$ of global CZ operators, extending andsharpening past results of Prats and Tolsa for the convolution case. Theweighted estimates we obtain, particularized to the Beurling operator on aLipschitz domain with normal to the boundary in the corresponding sharp Besovclass, may be used to deduce quantitative estimates for quasiregular mappingswith dilatation in the Sobolev space $W^{1,p}(\Omega)$, $p>2$.

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