Polya-Szego inequality and Dirichlet $p$-spectral gap for non-smooth spaces with Ricci curvature bounded below

TL;DR

This paper extends classical inequalities related to the Polya-Szego inequality and spectral gaps to non-smooth metric measure spaces with Ricci curvature bounds, providing new results and rigidity theorems in this generalized setting.

Contribution

It establishes a Polya-Szego inequality and sharp spectral gap results for the $p$-Laplace operator on $CD(K,N)$ spaces, extending classical results to non-smooth spaces.

Findings

Proved a Polya-Szego inequality in non-smooth $CD(K,N)$ spaces.

Established sharp spectral gap estimates for the $p$-Laplace operator.

Derived rigidity and almost rigidity results in $RCD(K,N)$ spaces.

Abstract

We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by $K>0$ and dimension bounded above by $N\in (1,\infty)$ in a synthetic sense, the so called $CD(K,N)$ spaces. We first establish a Polya-Szego type inequality stating that the $W^{1,p}$-Sobolev norm decreases under such a rearrangement and apply the result to show sharp spectral gap for the $p$-Laplace operator with Dirichlet boundary conditions (on open subsets), for every $p\in (1,\infty)$. This extends to the non-smooth setting a classical result of B\'erard-Meyer and Matei; remarkable examples of spaces fitting out framework and for which the results seem new include: measured-Gromov Hausdorff limits of Riemannian manifolds with Ricci$\geq K>0$, finite dimensional Alexandrov spaces with curvature$\geq K>0$, Finsler manifolds with Ricci$\geq K>0$. In the second part of the paper we prove new rigidity and almost rigidity results attached to the aforementioned inequalities, in the framework of $RCD(K,N)$ spaces, which seem original even for smooth Riemannian manifolds with Ricci$\geq K>0$.