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Gradient estimates for $(p,V)$-harmonic functions on Riemannian manifolds

Yuxin Dong, Hezi Lin, Weihao Zheng

TL;DR

This work extends gradient estimates to $(p,V)$-harmonic functions on complete Riemannian manifolds under Bakry-Émery curvature conditions. The authors develop a local gradient bound via a Moser iteration framework, supported by Laplacian/volume comparison and a Sobolev embedding tailored to $Ric_V\ge-(n-1)K$ and $|V|\le\theta$. They also obtain a global, explicit gradient bound for positive entire $(p,V)$-harmonic functions on complete noncompact manifolds, together with a Harnack-type consequence. The results generalize prior Li–Yau–type estimates to the $(p,V)$ setting and provide concrete quantitative control of $|\nabla u|/u$ in terms of geometric data and the drift vector field.

Abstract

In this paper, we study $(p,V)$-harmonic functions on complete Riemannian manifolds using the Moser iteration method. A volume comparison theorem and a Sobolev embedding theorem are established under the Bakry-$\acute{E}$mery curvature condition. Moreover, we obtain an explicit global gradient estimate for positive entire $(p,V)$-harmonic functions.

Gradient estimates for $(p,V)$-harmonic functions on Riemannian manifolds

TL;DR

This work extends gradient estimates to -harmonic functions on complete Riemannian manifolds under Bakry-Émery curvature conditions. The authors develop a local gradient bound via a Moser iteration framework, supported by Laplacian/volume comparison and a Sobolev embedding tailored to and . They also obtain a global, explicit gradient bound for positive entire -harmonic functions on complete noncompact manifolds, together with a Harnack-type consequence. The results generalize prior Li–Yau–type estimates to the setting and provide concrete quantitative control of in terms of geometric data and the drift vector field.

Abstract

In this paper, we study -harmonic functions on complete Riemannian manifolds using the Moser iteration method. A volume comparison theorem and a Sobolev embedding theorem are established under the Bakry-mery curvature condition. Moreover, we obtain an explicit global gradient estimate for positive entire -harmonic functions.

Paper Structure

This paper contains 7 sections, 15 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $(p,V)$-harmonic function and Bakry-$\bf\acute{E}$mery curvature
  4. Laplacian and volume comparison theorems
  5. Sobolev embedding theorem
  6. Local estimate
  7. Global explicit estimate

Key Result

Theorem 1.1

(WangZhang11) Let $(M^n,g)$ be a complete Riemannian manifold, and $o\in M$ be a fixed point. Assume the Ricci curvature satisfies $Ric\geq-(n-1)K$ on $B_o(2R)$, $K\geq0$. Suppose $u$ is a positive $p$-harmonic function, i.e., a positive solution to Then for any $x\in B_o(R)$, there exists a constant $C=C(n,p)$ such that

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • ...and 15 more