Bilinear Wavelet Representation of Calder\'on-Zygmund Forms

TL;DR

This paper introduces a novel wavelet-based representation of bilinear Calderón-Zygmund operators, enabling sharp weighted estimates and new Leibniz rules in Sobolev spaces, advancing the understanding of bilinear harmonic analysis.

Contribution

It provides a finite wavelet-based decomposition of bilinear Calderón-Zygmund operators, leading to improved weighted bounds and fractional differentiation rules.

Findings

Sparse $T(1)$-type bounds for bilinear operators

Sharp weighted bilinear estimates on Lebesgue and Sobolev spaces

New Leibniz-type rules in weighted Sobolev spaces

Abstract

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.