  We represent a bilinear Calder\'on-Zygmund operator at a given smoothness
level as a finite sum of cancellative, complexity zero operators, involving
smooth wavelet forms, and continuous paraproduct forms. This representation
results in a sparse $T(1)$-type bound, which in turn yields directly new sharp
weighted bilinear estimates on Lebesgue and Sobolev spaces. Moreover, we apply
the representation theorem to study fractional differentiation of bilinear
operators, establishing Leibniz-type rules in weighted Sobolev spaces which are
new even in the simplest case of the pointwise product.
