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# A Maximal Function Characterization of the Class $H^p$

D. L. Burkholder, R. F. Gundy and M. L. Silverstein
Transactions of the American Mathematical Society
Vol. 157 (Jun., 1971), pp. 137-153
DOI: 10.2307/1995838
Stable URL: http://www.jstor.org/stable/1995838
Page Count: 17
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## Abstract

Let $u$ be harmonic in the upper half-plane and $0 < p < \infty$. Then $u = \operatorname{Re} F$ 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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