Hp Theory on Compact Lie Groups

Abstract

The purpose of this article is to summarize some recent developments of H p spaces and their related theorems on a compact Lie group. I would like to thank Professor Sheng Gong for encouraging me to write this paper. Many results in this paper originally come from my Ph. D. thesis in Washington University, I would like to thank my thesis advisor, Professor Brian Blank for introducing me to this interesting topic. I also owe a great deal to Dr. Z. Xu, who provided me with a lot of materials for writing this paper. Since the article is going to present the theorems rather give the proofs, only part of the theorems will be sketched their proofs. Interested readers may see the listed references for further details and proofs. In some sections of the article, I will post some open questions to be solved. Atomic decomposition of Hardy spaces of real functions on Euclidean spaces first arose in the work of R. Coifman [Coi] and R. Laufm ter [L]. An abstract theory of atomic Hardy spaces was later developed by R. Coiman and G. Weiss [CW] in the context of spaces of homogeneous type. These spaces include Euclidean spaces and compact Lie groups but do not in general have the structure on which to base a theory of Hardy space defined by maximal functions. It was noted by Coifman and Weiss in [CW] and by Uchiyama in [U] that when a space of homogeneous type admits a certain family of kernels, a maximal function based Hardy space can be defined and shown equivalent to atomic Hardy space. Although the kernels in question are well-suited to an argument of L.