Discrete Schwarz rearrangement on lattice graphs

Abstract In this paper, we prove a discrete version of the generalized Riesz inequality on . 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 novel and self‐contained. In particular, we introduce a definition of the discrete rearrangement in higher dimensions. Moreover, we show that the method developed by Pruss for regular trees does not extend to . We solve a long‐standing open question raised by Alexander Pruss ( Duke Mathematical Journal , 1998, p. 494), and discussed with him in several communications in 2009–2010. Our method also provides a framework for proving other discrete rearrangement inequalities and opens the door to establishing existence of optimizers for many important discrete functional inequalities on , . We will also discuss severel applications of our results. To the best of our knowledge, our results are the first ones in the literature dealing with discrete rearrangement on , .