Wavelet representation of singular integral operators

This article develops a novel approach to the representation of singular integral operators Calderón–Zygmund type in terms continuous model operators, both classical and bi-parametric setting. The is realized as finite sum averages wavelet projections either cancellative or noncancellative type, which are themselves operators. Both properties out reach for established dyadic-probabilistic technique. Unlike their dyadic counterparts, our reflects additional kernel smoothness operator being analyzed. Our formulas lead naturally new family T(1) theorems on weighted Sobolev spaces whose index related smoothness. In one parameter case, we obtain space analogue \(A_2\) theorem; that is, sharp dependence norm T weight characteristic obtained full range exponents. setting, where local average sparse domination not generally available, quantitative \(A_p\) estimates best known, \(\max \{p,p'\}\ge 3\) fully case.