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 lnw 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-Émery curvature-dimension condition.

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