Date of Award
Doctor of Philosophy (PhD)
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.
Chair and Committee
Francesco Di Plinio, Loukas Grafakos, Gregory Knese, John McCarthy,
Stockdale, Cody B., "A Different Approach to Endpoint Weak-type Estimates for Calder̟n-Zygmund Operators" (2020). Arts & Sciences Electronic Theses and Dissertations. 2245.