On a Littlewood-Paley type inequality

Djordjević, Olivera; Pavlović, Miroslav

doi:10.1090/S0002-9939-07-09016-8

On a Littlewood-Paley type inequality
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by Olivera Djordjević and Miroslav Pavlović PDF

Proc. Amer. Math. Soc. 135 (2007), 3607-3611 Request permission

Abstract:

The following is proved: If $u$ is a function harmonic in the unit ball $B\subset \mathbb R^N$ and if $0<p\le 1,$ then the inequality \begin{equation*} \int _{\partial B}u^*(y)^p d\sigma \le C_{p,N}\left (|u(0)|^p +\int _B(1-|x|)^{p-1}|\nabla u(x)|^p dV(x)\right ) \end{equation*} holds, where $u^*$ is the nontangential maximal function of $u.$ This improves a recent result of Stoll. This inequality holds for polyharmonic and hyperbolically harmonic functions as well.

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