  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 $1\leq p \leq \infty$ 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.
