Global Stein theorem on Hardy spaces

AbstractLet\(f\) 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.

Source: European Journal of Applied Physiology - March 24, 2024 Category: Physiology Source Type: research