Global Stein theorem on Hardy spaces

Abstract

Let $f$ be an integrable function which has integral $0$ on $\mathbb{R}^n$. What is the largest condition on $|f|$ that guarantees that $f$ is in the Hardy space $\mathcal{H}^1(\mathbb{R}^n)$? When $f$ is compactly supported, it is well-known that the largest condition on $|f|$ is the fact that $|f|\in L \log L(\mathbb{R}^n)$. We consider the same kind of problem here, but without any condition on the support. We do so for $\mathcal{H}^1(\mathbb{R}^n)$, as well as for the Hardy space $\mathcal{H}_{\log}(\mathbb{R}^n)$ which appears in the study of pointwise products of functions in $\mathcal{H}^1(\mathbb{R}^n)$ and in its dual BMO.