International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 57-60, Pages 3129-3150
doi:10.1155/S0161171204402348
Extensions of the Hardy-Littlewood inequalities for Schwarz symmetrization
H. Hajaiej1
and C.A. Stuart2
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
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.