Optimal L^p-Riemannian Gagliardo-Nirenberg inequalities

TL;DR

This paper establishes the optimal L^p-Riemannian Gagliardo-Nirenberg inequalities on compact manifolds for 1 < p ≤ 2, extending previous Euclidean and Riemannian results within a unified framework.

Contribution

It proves the validity of these inequalities on compact Riemannian manifolds for a range of p, introducing a new distance lemma to unify Euclidean and Riemannian cases.

Findings

Proved optimal L^p-Riemannian Gagliardo-Nirenberg inequalities for 1 < p ≤ 2

Extended Euclidean inequalities to Riemannian manifolds

Developed a new distance lemma for the proof

Abstract

Let (M,g) be a compact Riemannian manifold of dimension n \geq 2. In this work we prove the validity of the optimal L^p-Riemannian Gagliardo-Nirenberg inequality for 1 < p \leq 2. Our proof relies strongly on a new distance lemma which. In particular, we extend L^p-Euclidean Gagliardo-Nirenberg inequalities due to Del Pino and Dolbeault and the optimal L^2-Riemannian Gagliardo-Nirenberg inequality due to Broutteland in a unified framework.