Discrete Schwarz rearrangement in lattice graphs

Abstract

In this paper, we prove a discrete version of the generalized Riesz inequality on ${\mathbb{Z}}^d$. As a consequence, we will derive the extended Hardy-Littlewood and Pólya-Szegö inequalities. We will also establish cases of equality in the latter. Our approach is totally novel and self-contained. In particular, we invented a definition for the discrete rearrangement in higher dimensions. Moreover, we show that the definition "suggested" by Pruss does not work. We solve a long-standing open question raised by Alexander Pruss in [Pru98, p494], Duke Math Journal, and discussed with him in several communications in 2009-2010, [Pru10]. Our method also provides a line of attack to prove other discrete rearrangement inequalities and opens the door to the establishment of optimizers of many important discrete functional inequalities in ${\mathbb{Z}}^d,$ $d\ge 2$. We will also discuss some applications of our findings. To the best of our knowledge, our results are the first ones in the literature dealing with discrete rearrangement on ${\mathbb{Z}}^d,$ $d\ge 2$.

We revised some expressions; All comments are welcome