Hardy Space Theory And Endpoint Estimates For Multi-Parameter Singular Radon Transforms

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]