Representations of functions harmonic in the upper half-plane and their applications

Abstract

In this thesis, new conditions for the validity of a generalized Poisson representation for a function harmonic in the upper half-plane have been found. These conditions differ from known ones by weaker growth restrictions inside the halfplane and stronger restrictions on the behavior on the real axis. We applied our results in order to obtain some new factorization theorems in Hardy and Nevanlinna classes. As another application we obtained a criterion of belonging to the Hardy class up to an exponential factor. Finally, our results allowed us to extend the Titchmarsh convolution theorem to linearly independent measures with unbounded support.