Characterizations of \(H^1\) and applications to singular integrals (Q1584489)

The author tries to solve the following problem. Problem: Find a necessary and sufficient size condition for an integrable compactly supported function on \(\mathbb{R}^n\) with mean zero to be in the Hardy space \(H^1\). The author obtains a satisfactory answer in the dimension one. Given a such function \(f\), define almost everywhere a function \(m_f\) on \(\mathbb{R}^1\) by setting \(m_f(a)=-\int_{\mathbb{R}} f(x)\log|x-a|dx\). He proves that \(f\in H^1(\mathbb{R}^1)\) if and only if \(m_f\) is a function of finite variation. The author also proves several equivalent characterizations of the Hardy space \(H^1\) on the unit sphere \(S^{n-1}\). As applications, he gives some new proofs for the \(L^p\) boundedness of singular integral operators and their truncated maximal operators.