ABP estimate on metric measure spaces via optimal transport

TL;DR

This paper develops a sharp ABP estimate on metric measure spaces with lower Ricci curvature bounds using optimal transport, extending classical results to non-smooth settings and offering new tools for geometric analysis.

Contribution

It introduces a novel ABP estimate in non-smooth metric measure spaces via optimal transport, bypassing Jacobi fields computation.

Findings

Establishes a sharp ABP estimate on metric measure spaces.

Proves new geometric and functional inequalities.

Provides a practical approach for non-smooth geometric analysis.

Abstract

By using optimal transport theory, we establish a sharp Alexandroff--Bakelman--Pucci (ABP) type estimate on metric measure spaces with synthetic Riemannian Ricci curvature lower bounds, and prove some geometric and functional inequalities including a functional ABP estimate. Our result not only extends the border of ABP estimate, but also provides an effective substitution of Jacobi fields computation in the non-smooth framework, which has potential applications to many problems in non-smooth geometric analysis.