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.
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