A quantitative second order estimate for (weighted) $p$-harmonic functions in manifolds under curvature-dimension condition

We build up a quantitative second order Sobolev estimate of $\ln w$ for positive $p$-harmonic functions $w$ in Riemannian manifolds under Ricci curvature bounded from blow and also for positive weighted $p$-harmonic functions $w$ in weighted manifolds under the Bakry-'{E}mery curvature-dimension condition.