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

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 ∈ (1 , ∞) 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 ∈ (1 , ∞). This extends to the non-smooth setting a classical result of Bérard-Meyer [14] and Matei [41] ; remarkable examples of spaces fitting our framework and for which the results seem new include: measured-Gromov Hausdorff limits of Riemannian manifolds with Ricci ≥ K > 0 , finite dimensional Alexandrov spaces with curvature ≥ K > 0 , Finsler manifolds with Ricci ≥ 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 are interesting even for smooth Riemannian manifolds with Ricci ≥ K > 0. [ABSTRACT FROM AUTHOR]