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

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 Bérard-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$\ge K>0$, finite dimensional Alexandrov spaces with curvature$\ge K>0$, Finsler manifolds with Ricci$\ge K>0$. 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$\ge K>0$.

arXiv comment33 pages. Final version published in Journal de Mathématiques Pures et Appliquées

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Cited by in corpus (9)

• Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds
• Quantitative Obata's Theorem
• Sharp Cheeger-Buser type inequalities in $\mathsf{RCD}(K,\infty )$ spaces
• A Talenti-type comparison theorem for $\mathrm{RCD}(K,N)$ spaces and applications
• The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces
• Sharp log-Sobolev inequalities in $\mathsf{CD}(0,N)$ spaces with applications
• The Pólya-Szegő inequality for smoothing rearrangements
• A reverse Hölder inequality for first eigenfunctions of the Dirichlet Laplacian on RCD(K,N) spaces
• Principal frequency of clamped plates on RCD(0,N) spaces: sharpness, rigidity and stability