Hardy Space Decompositions of $L^p(\mathbb{R}^n)$ for $0<p<1$ with Rational Approximation

TL;DR

This paper extends Hardy space decomposition results from one-dimensional functions to higher dimensions for $L^p(R^n)$ with $0<p<1$, using rational approximation techniques and analyzing the uniqueness of such decompositions.

Contribution

It generalizes the Hardy space decomposition from one dimension to higher dimensions and discusses the uniqueness of these decompositions.

Findings

Established higher-dimensional Hardy space decompositions for $L^p(R^n)$ with $0<p<1$

Proved the existence of such decompositions using rational approximation methods

Analyzed the conditions for the uniqueness of the decompositions.

Abstract

This paper aims to obtain decompositions of higher dimensional $L^p(\mathbb{R}^n)$ 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 \cite{DQ} recently obtained such Hardy space decomposition result: for any function $f\in L^p(\mathbb{R}),\ 0<p<1$, 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.