Global Stein theorem on Hardy spaces

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.