This work devises a new smooth representation formula for the compression of CZ operators on domains. As a first order consequence of this representation, we obtain a weighted, sharply quantified T(1)-type theorem on Sobolev spaces. Previous results of Prats and Prats-Tolsa are limited to unweighted bounds for convolution-type operators. Our weighted Sobolev inequalities are subsequently applied to obtain quantitative regularity results for solutions to the Beltrami equation with symbol in the critical class W^{k,2}(Omega). Alll past results, due to Prats among others, based on the Iwaniec scheme are of qualitative nature. Talk is based on current and ongoing joint work with Walton Green and Brett Wick (WUSTL)