A Talenti-type comparison theorem for the p-Laplacian on RCD(K,N) spaces and some applications

Abstract

In this paper, we prove a Talenti-type comparison theorem for the $p$-Laplacian with Dirichlet boundary conditions on open subsets of a normalized $\mathrm{RCD}(K,N)$ space with $K>0$ and $N\in (1,\infty )$. The obtained Talenti-type comparison theorem is sharp, rigid and stable with respect to measured Gromov–Hausdorff topology. As an application of such Talenti-type comparison, we establish a sharp and rigid reverse Hölder inequality for first eigenfunctions of the $p$-Laplacian and a related quantitative stability result.