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Discrete Littlewood-Paley-Stein Theory And Wolff Potentials On Homogeneous Spaces And Multi-Parameter Hardy Spaces, Yayuan Xiao

Wayne State University Dissertations

This dissertation consists of two parts:

In part I, We establish a new atomic decomposition of the multi-parameter Hardy spaces of homogeneous type and obtain the associated $H^p-L^p$ and $H^p-H^p$ boundedness criterions for singular integral operators. On the other hand, we compare the Wolff and Riesz potentials on spaces of homogenous type, followed by a Hardy-Littlewood-Sobolev type inequality. Then we drive integrability estimates of positive solutions to the Lane-Emden type integral systems on spaces of homogeneous type.

In part II, We establish a $(p,2)$-atomic decomposition of the Hardy space associated with different homogeneities for $0