A Talenti-type comparison theorem for the $p$-Laplacian on $\mathrm{R}\mathrm{C}\mathrm{D}(K,N)$ spaces and some applications

In this paper, we prove a Talenti-type comparison theorem for the $p$-Laplacian with Dirichlet boundary conditions on open subsets of a $\mathrm{R}\mathrm{C}\mathrm{D}(K,N)$ space with $K>0$ and $N\in (1,\mathrm{\infty })$. 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.