Second-order estimates for the p-Laplacian in RCD spaces

We establish quantitative second-order Sobolev regularity for functions having a 2-integrable p-Laplacian in bounded RCD spaces, with p in a suitable range. In the finite-dimensional case, we also obtain Lipschitz regularity under the assumption that p-Laplacian is sufficiently integrable. Our results cover both p-Laplacian eigenfunctions and p-harmonic functions having relatively compact level sets.