Global Stein Theorem on Hardy spaces

TL;DR

This paper investigates the conditions on the decay of integrable functions at infinity that ensure their membership in Hardy spaces, extending classical results and exploring implications for function products and dual spaces.

Contribution

It establishes a comprehensive criterion for functions with integral zero to belong to Hardy spaces based on their behavior at infinity, generalizing known compact support results.

Findings

Characterization of Hardy space membership via decay conditions at infinity

Extension of L log L conditions to unbounded functions

Implications for products of Hardy space functions and BMO duality

Abstract

Let f be an integrable function which has integral 0 on R n. What is the largest condition on |f | that guarantees that f is in the Hardy space H 1 (R n)? When f is compactly supported, it is well-known that it is necessary and sufficient that |f | belongs to L log L(R n). We are interested here in conditions at $\infty$. We do so for H 1 (R n), as well as for the Hardy space H log (R n) which appears in the study of pointwise products of functions in H 1 (R n) and in its dual BMO.