Weighted Banach spaces of holomorphic functions in the upper half plane

TL;DR

This paper studies weighted Banach spaces of holomorphic functions in the upper half-plane, providing conditions for their non-triviality and analyzing the asymptotic behavior of functions within these spaces.

Contribution

It offers a complete characterization of when these weighted Banach spaces are non-trivial based on properties of the weight functions.

Findings

Characterization of non-trivial weighted Banach spaces

Necessary and sufficient conditions for space triviality

Analysis of asymptotic behavior of functions in these spaces

Abstract

We consider weighted banach spaces of holomorphic functions on the upper half plane that are determined by $ \|f\|=\sup_{y>0,-\infty<x<\infty}p(y)|f(x+iy)|<\infty $ for a very large class of weight functions p. We completely solve the problem whether such banach spaces are trivial or not by giving necessary and sufficient conditions stated in terms of some simple properties of the weight function. Further, we investigate the behaviour at infinity of some functions that belong to some of the banach spaces under consideration.