Date of Award

Winter 12-15-2021

Author's School

Graduate School of Arts and Sciences

Author's Department

Mathematics

Degree Name

Doctor of Philosophy (PhD)

Degree Type

Dissertation

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.

Language

English (en)

Chair and Committee

Brett Wick

Committee Members

Francesco Di Plinio

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