Boundary values of holomorphic functions and heat kernel method in translation-invariant distribution spaces

We study boundary values of holomorphic functions in translation-invariant distribution spaces of type D-E' . New edge of the wedge theorems are obtained. The results are then applied to represent D-E' as a quotient space of holomorphic functions. We also give representations of elements D-E' of via the heat kernel method. Our results cover as particular instances the cases of boundary values, analytic representations and heat kernel representations in the context D-LP' , B' of the Schwartz spaces and their weighted versions.