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arXiv · 2209.01003

Discrete Schwarz rearrangement in lattice graphs

Hichem Hajaiej·Fengwen Han·Bobo Hua
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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 $\mathbb{Z}^d,$ $d\geq2$.

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Hichem Hajaiej, Fengwen Han, Bobo Hua. 2022-09-02. Discrete Schwarz rearrangement in lattice graphs. https://arxiv.org/abs/2209.01003

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