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

Jiayin Liu, Shijin Zhang, Yuan Zhou

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-Émery curvature-dimension condition.

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

Abstract

We build up a quantitative second order Sobolev estimate of for positive -harmonic functions in Riemannian manifolds under Ricci curvature bounded from blow and also for positive weighted -harmonic functions in weighted manifolds under the Bakry-Émery curvature-dimension condition.
Paper Structure (4 sections, 10 theorems, 121 equations)

This paper contains 4 sections, 10 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm2']}
  4. Proof of Lemma \ref{['d1983']}

Key Result

Theorem 1.1

Suppose that $(M^n,g)$ satisfies $Ric_g \ge - \kappa$ for some $\kappa\ge 0$. Let $1<p<\infty$ and ${\gamma}<3+ \frac{p-1}{n-1}$. For any positive $p$-harmonic function w in a domain $\Omega\subset M$, we have $|\nabla \ln w|^{\frac{p-{\gamma}}{2}}\nabla \ln w\in W^{1,2}_{\mathop\mathrm{\,loc\,}}(\ whenever $B(z,4r)\Subset \Omega$. In particular, if $1<p<3+\frac{2}{n-2}$, then $\nabla^2\ln w\in L

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • proof : Proof of Lemma \ref{['divp-gz0']}
  • Lemma 3.4
  • ...and 9 more