$L^2$-Caffarelli--Kohn--Nirenberg inequalities on metric measure spaces

TL;DR

This paper extends $L^2$-Caffarelli--Kohn--Nirenberg inequalities to metric measure spaces, revealing geometric implications like reverse volume comparison and characterizing volume cones, with stability results on these spaces.

Contribution

It introduces a unified framework for $L^2$-CKN inequalities on metric measure spaces, linking inequalities to geometric properties and characterizing volume cones.

Findings

Inequalities imply reverse volume comparison of G"unther type.

On measure contraction spaces, inequalities hold iff the space is a volume cone.

Stability results are established for inequalities on volume cones.

Abstract

Motivated by the sharp constants in the $L^2$-Caffarelli--Kohn--Nirenberg (or $L^2$-CKN for short) inequalities on Euclidean spaces, we study, in a unified framework, a sequence of $L^2$-CKN inequalities on metric measure spaces. On a general metric measure space, this sequence implies a reverse volume comparison of G\"unther type. Moreover, on a subclass of spaces admitting the measure contraction property, we show that this sequence of $L^2$-CKN inequalities are valid if and only if the spaces are volume cones. We also provide a stability result for inequalities of this type on volume cones.