Weighted Sobolev Inequalities in CD(0,N) spaces

TL;DR

This paper extends weighted Sobolev inequalities to non-smooth CD(0,N) spaces, enabling bounds on heat kernels and broadening the scope of geometric analysis beyond Riemannian manifolds.

Contribution

It establishes global weighted Sobolev inequalities in non-compact CD(0,N) spaces, generalizing previous Riemannian results to more general metric measure spaces.

Findings

Weighted Sobolev inequalities hold in non-compact CD(0,N) spaces.

Derived uniform bounds for heat kernels using weighted Nash inequalities.

Extended classical geometric analysis results to non-smooth spaces.

Abstract

In this note, we prove global weighted Sobolev inequalities on non-compact CD(0,N) spaces satisfying a suitable growth condition, extending to possibly non-smooth and non-Riemannian structures a previous result by V. Minerbe stated for Riemannian manifolds with non-negative Ricci curvature. We use this result in the context of RCD(0,N) spaces to get a uniform bound of the corresponding weighted heat kernel via a weighted Nash inequality.