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

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