A reverse H\"older inequality for first eigenfunctions of the Dirichlet Laplacian on RCD(K,N) spaces

In the framework of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by a positive constant in a synthetic sense, we establish a sharp and rigid reverse-H\"older inequality for first eigenfunctions of the Dirichlet Laplacian. This generalises to the positively curved and non-smooth setting the classical "Chiti Comparison Theorem". We also prove a related quantitative stability result which seems to be new even for smooth Riemannian manifolds.Comment: 14 pages. Final Version, to appear in the Proc. Amer. Math. So