Products of Functions in Hardy and Lipschitz or Bmo Spaces

TL;DR

This paper investigates the product of functions in Hardy spaces and their duals, establishing how such products can be expressed as sums involving Hardy-Orlicz spaces or Hardy spaces, depending on parameters.

Contribution

It introduces a distributional definition for products of Hardy space functions with dual space functions and characterizes their decomposition into sums within Hardy-Orlicz or Hardy spaces.

Findings

Product bxh can be written as a sum of an integrable function and a Hardy-Orlicz distribution.

The decomposition depends on the value of p in Hardy spaces.

Provides a new framework for understanding products in Hardy and related spaces.

Abstract

We define as a distribution the product of a function (or distribution) h in some Hardy space Hp with a function b in the dual space of Hp. Moreover, we prove that the product bxh may be written as the sum of an integrable function with a distribution that belongs to some Hardy-Orlicz space, or to the same Hardy space Hp, depending on the values of p.