Bulletin of the Brazilian Mathematical Society

, Volume 34, Issue 2, pp 211–217

A Littlewood-Paley type inequality

Original Paper

DOI: 10.1007/s00574-003-0008-1

Cite this article as:
Stević, S. Bull Braz Math Soc (2003) 34: 211. doi:10.1007/s00574-003-0008-1

Abstract.

In this note we prove the following theorem: Let u be a harmonic function in the unit ball \( B \subset {\mathbf{R}}^{n} \) and \( p \in {\left[ {\frac{{n - 2}} {{n - 1}},1} \right]} \). Then there is a constant C = C(p, n) such that
$$ {\mathop {\sup }\limits_{0 \leqslant r < 1} }{\kern 1pt} {\kern 1pt} {\int_S {{\left| {u{\left( {r\zeta } \right)}} \right|}^{p} d\sigma {\left( \zeta \right)} \leqslant C{\left( {{\left| {u{\left( 0 \right)}} \right|}^{p} + {\int_B {{\left| {\nabla u{\left( x \right)}} \right|}^{p} {\left( {1 - {\left| x \right|}} \right)}^{{p - 1}} dV{\left( x \right)}} }} \right)}} } $$
.

Keywords:

Harmonic functionsLittlewood-Paley inequalityHardy spacemaximal functionunit ball

Mathematical subject classification:

31B05

Copyright information

© Sociedade Brasileira de Matemática 2003

Authors and Affiliations

  1. 1.Matematički FakultetBeogradSERBIA