Hardy space decompositions of L^p(ℝⁿ) for 0 < p < 1 with rational approximation

Abstract

This paper aims to obtain decompositions of higher dimensional L p (R n ) functions into sums of non-tangential boundary limits of the corresponding Hardy space functions on tubes for the index range 0 < p < 1. In the one-dimensional case, Deng and Qian [5] recently obtained such Hardy space decomposition result: for any function f ∈ L p (R), 0 < p < 1, there exist functions f 1 and f 2 such that f = f 1 + f 2 , where f 1 and f 2 are, respectively, the non-tangential boundary limits of some Hardy space functions in the upper-half and lower-half planes. In the present paper, we generalize the one-dimensional Hardy space decomposition result to the higher dimensions, and discuss the uniqueness issue of such decomposition.

Composition Operators on Dirichlet Spaces over the Half-plane

“…For more general case, what kind of the rational function is belong to the Dirichlet space D(C + )? The second author and her coworkers [13] have considered this type question for Hardy spaces over tube domains, and obtain that the analytic rational functions with L p non-tangent boundary limit functions are belong to relative Hardy spaces. How about the Dirichlet spaces?…”