Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems

TL;DR

This paper investigates the existence of maximizers for generalized Gagliardo-Nirenberg inequalities and a new Riesz energy inequality, employing advanced lemmas to extend previous results in non-local analysis.

Contribution

It introduces new existence results for maximizers in non-local inequalities using generalized lemmas and extends classical inequalities to broader contexts.

Findings

Existence of maximizers for generalized Gagliardo-Nirenberg inequalities.

Existence of maximizers for a new Riesz energy inequality.

Development of generalized Lieb's Translation Lemma and Brézis–Lieb lemma.

Abstract

In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo-Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Lieb's Translation Lemma and a Riesz energy version of the Br\'ezis--Lieb lemma.