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