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Transactions of the American Mathematical Society

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Extended Hardy-Littlewood inequalities and some applications

Author: Hichem Hajaiej
Journal: Trans. Amer. Math. Soc. 357 (2005), 4885-4896
MSC (2000): Primary 26D15
Published electronically: July 19, 2005
MathSciNet review: 2165392
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Abstract: We establish conditions under which the extended Hardy-Little- wood inequality

\begin{displaymath}\int \limits_{\mathbb{R} ^N} H\big{(}\vert x\vert,\, u_1(x), ... ...ig{(}\vert x\vert,\, u_1^*(x), \ldots , u_m^*(x)\big{)}\, dx, \end{displaymath}

where each $u_i$ is non-negative and $u_i^*$ denotes its Schwarz symmetrization, holds. We also determine appropriate monotonicity assumptions on $H$ such that equality occurs in the above inequality if and only if each $u_i$ is Schwarz symmetric. We end this paper with some applications of our results in the calculus of variations and partial differential equations.

References [Enhancements On Off] (What's this?)

  • 1. Hajaiej H., Cases of Equality and Strict Inequality in the Extended Hardy-Littlewood Inequalities, Proc. Roy. Soc. Edinburgh, 135A (2005), 643-661.
  • 2. Hajaiej H., Stuart C. A., Extensions of the Hardy-Littlewood Inequalities for Schwarz Symmetrization, Int. J. Math. Math. Sci. 2004, 3129-3150. MR 2110793
  • 3. Hajaiej H., Stuart C. A., Existence and Non-existence of Schwarz Symmetric Ground States for Elliptic Eigenvalue Problems, Matematica pura ed Applicata, 186, electronic, 2004.
  • 4. Folland G.B., Real Analysis: Modern Techniques and their Applications, Pure and Applied Mathematics, John Wiley and Sons, New York, 1999. MR 1681462 (2000c:00001)
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Additional Information

Hichem Hajaiej
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904-4137

Received by editor(s): January 13, 2004
Published electronically: July 19, 2005
Dedicated: Dedicated to my Mother: To you Omi
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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