A maximal function characterization of the class $H^{p}$

Abstract:

Let $u$ be harmonic in the upper half-plane and $0 < p < \infty$. Then $u = \text {Re} F$ for some analytic function $F$ of the Hardy class ${H^p}$ if and only if the nontangential maximal function of $u$ is in ${L^p}$. A general integral inequality between the nontangential maximal function of $u$ and that of its conjugate function is established.