Note on vector valued Hardy spaces related to analytic functions having distributional boundary values

Analytic functions defined on a tube domain $T^C\subset {\mathbb{C}}^n$ and taking values in a Banach space $X$ which are known to have $X$-valued distributional boundary values are shown to be in the Hardy space $H^p(T^C,X)$ if the boundary value is in the vector valued Lebesgue space $L^p({\mathbb{R}}^n,X)$, where $1\le p\le \mathrm{\infty }$ and $C$ is a regular open convex cone. Poisson integral transform representations of elements of $H^p(T^C,X)$ are also obtained for certain classes of Banach spaces, including reflexive Banach spaces.