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arXiv · 2009.01212

Wavelet Representation of Singular Integral Operators

Francesco Di Plinio·Brett D. Wick·Tyler Williams
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Abstract

This article develops a novel approach to the representation of singular integral operators of Calderón-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 Calderón-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.

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BibTeXRIS

Francesco Di Plinio, Brett D. Wick, Tyler Williams. 2021-01-04. Wavelet Representation of Singular Integral Operators. https://arxiv.org/abs/2009.01212

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