Factorization of some Hardy type spaces of holomorphic functions

TL;DR

This paper characterizes the factorization of Hardy type spaces of holomorphic functions in the upper half-plane, showing that functions in these spaces can be expressed as products of functions from Hardy space and its dual, extending known results from the unit disc.

Contribution

It generalizes the factorization characterization of Hardy spaces from the unit disc to the upper half-plane, providing a new understanding of these spaces.

Findings

Product of functions in Hardy space and its dual belongs to a Hardy type space

Every function in this Hardy type space can be factored into such a product

Extends classical results from the unit disc to the upper half-plane

Abstract

We prove that the pointwise product of two holomorphic functions of the upper half-plane, one in the Hardy space $\mathcal H^1$, the other one in its dual, belongs to a Hardy type space. Conversely, every holomorphic function in this space can be written as such a product. This generalizes previous characterization in the context of the unit disc.