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ISSN 1088-6850(e) ISSN 0002-9947(p)

Extended Hardy-Littlewood inequalities and some applications

Author(s): Hichem Hajaiej
Journal: Trans. Amer. Math. Soc. 357 (2005), 4885-4896.
MSC (2000): Primary 26D15
Posted: 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

is non-negative and

denotes its Schwarz symmetrization, holds. We also determine appropriate monotonicity assumptions on

such that equality occurs in the above inequality if and only if each

is Schwarz symmetric. We end this paper with some applications of our results in the calculus of variations and partial differential equations.

References:

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)
5.: Tahraoui R., Symmetrization Inequalities, Nonlinear Analysis TMA, 27 (1996), pp. 933-955, Corrigendum 33 (2000), 535. MR 1404592 (97h:35008); MR 1725394

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