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

TL;DR

This paper develops a quantitative second order Sobolev estimate for positive p-harmonic functions in Riemannian and weighted manifolds, extending classical results under curvature-dimension conditions.

Contribution

It introduces a new second order Sobolev estimate for $\, ext{ln}\, w$ of positive p-harmonic functions in manifolds with Ricci or Bakry-Émery curvature bounds.

Findings

Established second order Sobolev estimates for p-harmonic functions.

Extended estimates to weighted manifolds under curvature-dimension conditions.

Provided tools for further geometric analysis of p-harmonic functions.

Abstract

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.