Wavelet Representation of Singular Integral Operators

TL;DR

This paper introduces a new wavelet-based representation for Calderón-Zygmund singular integral operators, capturing kernel smoothness and enabling sharp weighted norm estimates in Sobolev spaces, including bi-parametric cases.

Contribution

It develops a novel wavelet representation that reflects kernel smoothness and leads to new $T(1)$ theorems and sharp weighted estimates, surpassing dyadic-probabilistic methods.

Findings

Wavelet representation captures kernel smoothness.

Sharp $A_2$ theorem for Sobolev spaces.

Quantitative $A_p$ estimates in bi-parametric setting.

Abstract

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.