Characterizations of $H^{1}$ and applications to singular integrals

Abstract

We give a necessary and sufficient condition for an integrable compactly supported function with mean value zero on the line to be in the Hardy space $H^{1}(\mathbf{R}^{1})$. As a corollary, we obtain a new characterization of $H^{1}(\mathbf{S}^{1})$ and $p$ independence of the spectrum of homogeneous Calderón-Zygmund operators.