  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.
