Extended Hardy-Littlewood inequalities and some applications

Abstract:

We establish conditions under which the extended Hardy-Little- wood inequality \begin{equation*} \int \limits _{\mathbb {R}^N} H\big {(}|x|, u_1(x), \ldots , u_m(x)\big {)} dx \leq \int \limits _{\mathbb {R}^N} H\big {(}|x|, u_1^*(x), \ldots , u_m^*(x)\big {)} dx, \end{equation*} 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.