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

Burkholder, D. L.; Gundy, R. F.; Silverstein, M. L.

doi:10.1090/S0002-9947-1971-0274767-6

A maximal function characterization of the class $H^{p}$
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by D. L. Burkholder, R. F. Gundy and M. L. Silverstein PDF

Trans. Amer. Math. Soc. 157 (1971), 137-153 Request permission

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.

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A note on analytic functions in the unit circle

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