Abstract

The aim of this thesis is to investigate weak-type inequalities for linear and multilinear Calderón-Zygmund operators in Euclidean and weighted settings using the Calderón- Zygmund decomposition and ideas inspired by Nazarov, Treil, and Volberg. In the linear setting, a new simple proof of the classical weak-type (1; 1) property is given with motivation For multilinear Calderón-Zygmund operators, the Nazarov-Treil-Volberg ideas lead to a new proof of the weak-type (1,. . . ,1; 1/m) estimate. Connecting the weighted and multilinear settings, a weighted weak-type estimate for multilinear Calderón-Zygmund operators is proved. Two proofs for the weighted multilinear inequality are presented – one proof uses the Calderón-Zygmund decomposition, and the other proof uses ideas inspired by Nazarov, Treil, and Volberg. Additionally, a weak-type (q; q) estimate is proved for Calderón-Zygmund operators whose kernels satisfy an Lq(Rn)-adapted integral smoothness condition, weaker than is typically assumed. Two proofs of the weak-type (q; q) result are presented – one uses the Calderón-Zygmund decomposition and the other is inspired by Nazarov, Treil, and Volberg. Finally, the Nazarov-Treil-Volberg method is used to investigate the dimensional dependence of the weak-type (1; 1) norm of the Riesz transforms. Denoting the jth Riesz transform on Rn by Rj , we show that the weak-type (1; 1) norm of Rj grows at most as a constant times log n.

Committee Chair

Brett Wick

Committee Members

Francesco Di Plinio, Loukas Grafakos, Gregory Knese, John McCarthy,

Degree

Doctor of Philosophy (PhD)

Author's Department

Mathematics

Author's School

Graduate School of Arts and Sciences

Document Type

Dissertation

Date of Award

Spring 5-15-2020

Language

English (en)

Author's ORCID

http://orcid.org/0000-0003-1020-2418

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Mathematics Commons

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