Discrete Schwarz rearrangement in lattice graphs

TL;DR

This paper introduces a novel discrete rearrangement concept on lattice graphs, proving a generalized Riesz inequality and related inequalities, solving a long-standing open problem, and opening new avenues for discrete functional analysis.

Contribution

It presents the first definition of discrete rearrangement in higher dimensions and proves a generalized Riesz inequality on 5^d, addressing a long-standing open question.

Findings

Established a discrete version of the generalized Riesz inequality on 5^d

Derived extended Hardy-Littlewood and Pf3lya-Szegf6 inequalities

Solved a long-standing open problem in discrete rearrangement theory

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 P\'olya-Szeg\"o 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 $\mathbb{Z}^d,$ $d\geq2$. 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 $\mathbb{Z}^d,$ $d\geq2$.