Abstract

This thesis 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, this new representation reflects the additional kernel smoothness of the operator being analyzed. These 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, the Sobolev space analogue of the $A_2$ theorem is proven; 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, quantitative $A_p$ estimates are established which are best known, and sharp in the range $\max\{p,p'\}\geq 3$ for the fully cancellative case.

Committee Chair

Brett Wick

Committee Members

Francesco Di Plinio

Degree

Doctor of Philosophy (PhD)

Author's Department

Mathematics

Author's School

Graduate School of Arts and Sciences

Document Type

Dissertation

Date of Award

Winter 12-15-2021

Language

English (en)

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Mathematics Commons

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