Access Type

Open Access Embargo

Date of Award

January 2018

Degree Type

Dissertation

Degree Name

Ph.D.

Department

Mathematics

First Advisor

Guozhen Lu

Second Advisor

Peiyong Wang

Abstract

In [12], Christ, Nagel, Stein and Waigner studied the L p theories for the singular Radon Trans-

forms. Furthermore, B. Street in [68], and Stein and Street in [64–67] extended the theories of the

L p boundedness for multi-parameter singular integral operators, such as the Calderón Zygmund

operators and singular Radon transforms. In this dissertation, we will study the Hardy space H p

and its dual space associated with both the one-parameter and multi-parameter singular Radon

transforms, and consider the boundedness of the singular Radon transforms on such Hardy spaces

H p when 0 ≤ p ≤ 1.

Inspired by recent characterization of the Hardy spaces on product spaces, we will take advan-

tage of the discrete Littlewood-Paley analysis [14,32,43] to define the Hardy spaces H p and the

Carleson measure spaces CMO p associated with the multi-parameter singular Radon transforms.

Moreover, we will prove the H p boundedness of those operators and thus obtain the endpoint es-

timates for the L p boundedness of the singular Radon transforms by Christ, Nagel, Wainger and

Stein [12] and and for multi-parameter singular Radon transforms by Street and Stein [65–68].

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