Dominating manifolds by radial spaces

TL;DR

This paper develops a rearrangement technique for functions on CD(k,n)-spaces to establish a Moser-Trudinger inequality, characterizing manifolds with lower Ricci bounds that support such inequalities.

Contribution

It introduces a novel rearrangement method on CD(k,n)-spaces and applies it to prove a Moser-Trudinger inequality in this setting, providing new geometric insights.

Findings

Rearrangement technique valid on CD(k,n)-spaces

Moser-Trudinger inequality established for these spaces

Characterization of manifolds with Ricci bounds supporting the inequality

Abstract

This paper is devoted to a kind of rearrangement of functions on CD(k,n)-spaces, which satisfy a Polya-Szeg\"o type inequality. We use this rearrangement to prove the validity of a Moser-Trudinger type inequality on a wide class of metric measure spaces satisfying a CD(k,n)-curvature-dimension inequality. As a consequence, we give a characterization, among manifolds with lower bounded Ricci curvature, of those admitting a Moser-Trudinger type inequality.