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

TL;DR

This paper explores vector valued Hardy spaces in tube domains, establishing their boundary value properties and providing a Poisson integral representation for such functions.

Contribution

It defines vector valued Hardy spaces in tube domains and proves boundary value theorems for functions with distributional boundary values in Hilbert spaces.

Findings

Vector valued Hardy spaces are well-defined in tube domains.

Boundary values in L^p imply membership in Hardy spaces.

A Poisson integral representation for these functions is established.

Abstract

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.