The Gagliardo-Nirenberg inequality on metric measure spaces

TL;DR

This paper establishes a link between the Gagliardo-Nirenberg inequality and volume growth in metric measure spaces, with applications to Finsler geometry and smooth metric measure spaces, revealing geometric implications of functional inequalities.

Contribution

It proves that satisfying the Gagliardo-Nirenberg inequality with a specific exponent characterizes the volume growth in metric measure spaces and applies this to Finsler and smooth metric measure spaces.

Findings

Spaces with the inequality have exactly n-dimensional volume growth

Finsler manifolds with the inequality and nonnegative Ricci curvature have zero flag curvature

Provides an alternative proof for a key result in smooth metric measure spaces

Abstract

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Gagliardo-Nirenberg inequality with the same exponent $n$ $(n\geq 2)$, then it has exactly the $n$-dimensional volume growth. Besides, two interesting applications have also been given. The one is that we show that if a complete $n$-dimensional Finsler manifold of nonnegative $n$-Ricci curvature satisfies the Gagliardo-Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. The other one is that we give an alternative proof to Mao's main result in [23] for smooth metric measure spaces with nonnegative weighted Ricci curvature.