Vector valued Hardy spaces related to analytic functions having distributional boundary values

The Hardy space $H^{p}$ of vector valued analytic functions in tube domains in $\mathbb{C}^{n}$ and with values in Banach space are defined. Vector valued analytic functions in tube domains in $\mathbb{C}^{n}$ with values in Hilbert space and which have vector valued tempered distributions as boundary value are proved to be in $H^{p}$ corresponding to Hilbert space if the boundary value is in $L^{p}$ with values in Hilbert space. A Poisson integral representation for such vector valued analytic functions is obtained.