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

In this paper we prove an exponential covering lemma implying the three dimensional case of a well-known conjecture formulated by A. Zygmund circa 1935 and solved Córdoba in 1978. Our approach avoids subtle argument involving power series function.