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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: July 2, 2007
MathSciNet review: 2336576
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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

$\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 $ 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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Additional Information

Olivera Djordjevic
Affiliation: Fakultet organizacionih nauka, Jove Ilića 154, Belgrade, Serbia
Email: oliveradj@fon.bg.ac.yu

Miroslav Pavlovic
Affiliation: Matematički fakultet, Studentski trg 16, Belgrade, Serbia
Email: pavlovic@matf.bg.ac.yu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09016-8
PII: S 0002-9939(07)09016-8
Keywords: Littlewood-Paley inequalities, harmonic functions in $\mathbb R^N$
Received by editor(s): August 18, 2006
Published electronically: July 2, 2007
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2007 American Mathematical Society