A reverse H\"older inequality for first eigenfunctions of the Dirichlet Laplacian on RCD(K,N) spaces

TL;DR

This paper extends the classical Chiti Comparison Theorem to non-smooth metric measure spaces with Ricci curvature bounds, establishing a sharp reverse-H"older inequality for first eigenfunctions of the Dirichlet Laplacian, along with a new stability result.

Contribution

It introduces a sharp, rigid reverse-H"older inequality for eigenfunctions in non-smooth spaces with positive Ricci curvature bounds, generalizing classical results.

Findings

Proves a sharp reverse-H"older inequality for eigenfunctions.

Establishes a quantitative stability result.

Generalizes classical comparison theorems to non-smooth settings.

Abstract

In the framework of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by a positive constant in a synthetic sense, we establish a sharp and rigid reverse-H\"older inequality for first eigenfunctions of the Dirichlet Laplacian. This generalises to the positively curved and non-smooth setting the classical "Chiti Comparison Theorem". We also prove a related quantitative stability result which seems to be new even for smooth Riemannian manifolds.