International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 59, Pages 3129-3150
doi:10.1155/S0161171204402348
Extensions of the Hardy-Littlewood inequalities for Schwarz symmetrization
1Department of Mathematics, University of Virginia, Charlottesville 22902, VA, USA
2Institut d'Analyse et Calcul Scientifique (IACS), Faculte des Sciences de Base (FSB), École Polytechnique Fédérale de Lausanne (EPFL), Lausanne 1015, Switzerland
Received 4 February 2004
Copyright © 2004 H. Hajaiej and C. A. Stuart. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
For a class of functions H:(0,∞)×ℝ+2→ℝ, including discontinuous functions of Carathéodory type, we establish that ∫ℝNH(|x|,u(x),v(x))dx≤∫ℝNH(|x|,u*(x),v*(x))dx, where u*(x) and v*(x) denote the Schwarz symmetrizations of nonnegative functions u and v.