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

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}$),
0<p<1$,thereexistfunctions$f_1$and$f_2$suchthat$f=f_1+f_2$,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.