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Hardy type inequalities on manifolds with nonnegative Ricci curvature

Yuxin Dong, Hezi Lin, Lingen Lu

Abstract

We prove the Heisenberg-Pauli-Weyl inequality, Hardy-Sobolev inequality, and Caffarelli-Kohn-Nirenberg (CKN) inequality on manifolds with nonnegative Ricci curvature and Euclidean volume growth, of dimension n>=3.

Hardy type inequalities on manifolds with nonnegative Ricci curvature

Abstract

We prove the Heisenberg-Pauli-Weyl inequality, Hardy-Sobolev inequality, and Caffarelli-Kohn-Nirenberg (CKN) inequality on manifolds with nonnegative Ricci curvature and Euclidean volume growth, of dimension n>=3.
Paper Structure (4 sections, 7 theorems, 27 equations)

This paper contains 4 sections, 7 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hardy inequality and Heisenberg-Pauli-Weyl inequality
  4. Hardy-Sobolev inequality and CKN inequality

Key Result

Theorem 2.1

BK Let $(M^n,g)$ be a complete noncompact Riemannian manifold with nonnegative Ricci curvature and $u:M\to\mathbb{R}$ be a fast decaying function such that $|Du|\in L^p(M),p>1$. Then one has where $\theta$ denote the asymptotic volume ratio of $M$.

Theorems & Definitions (13)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • ...and 3 more