On a Littlewood-Paley type inequality

Djordjević, Olivera; Pavlović, Miroslav

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

On a Littlewood-Paley type inequality

Authors: Olivera Djordjevic and Miroslav Pavlovic
Journal: Proc. Amer. Math. Soc. 135 (2007), 3607-3611
MSC (2000): Primary 31B05
DOI: https://doi.org/10.1090/S0002-9939-07-09016-8
Published electronically: July 2, 2007
MathSciNet review: 2336576
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Abstract: The following is proved: If is a function harmonic in the unit ball $B\subset \mathbb{R}^N$ and if $0<p\le 1,$ then the inequality

$\displaystyle \int_{\partial B}u^*(y)^p\,d\sigma\le C_{p,N}\left(\vert u(0)\vert^p +\int_B(1-\vert x\vert)^{p-1}\vert\nabla u(x)\vert^p\,dV(x)\right)$

holds, where

is the nontangential maximal function of

This improves a recent result of Stoll. This inequality holds for polyharmonic and hyperbolically harmonic functions as well.

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