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Some Hardy and Rellich type inequalities for affine connections

Pengyan Wang, Huiting Chang

Abstract

In this article we study various forms of the Hardy inequality for affine connections on a complete noncompact Riemannian manifold, including the two-weight Hardy inequality, the improved Hardy inequality, the Rellich inequality, the Hardy-Poincaré inequality and the Heisenberg-Pauli-Weyl inequality. Our results improve and include many previously known results as special cases.

Some Hardy and Rellich type inequalities for affine connections

Abstract

In this article we study various forms of the Hardy inequality for affine connections on a complete noncompact Riemannian manifold, including the two-weight Hardy inequality, the improved Hardy inequality, the Rellich inequality, the Hardy-Poincaré inequality and the Heisenberg-Pauli-Weyl inequality. Our results improve and include many previously known results as special cases.
Paper Structure (7 sections, 14 theorems, 168 equations)

This paper contains 7 sections, 14 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Torsion-free affine connections $D^{\lambda,\mu}$
  4. Affine gradient, affine Laplacian, affine divergence and affine Ricci curvature.
  5. Hardy type inequalities and their proofs
  6. Rellich-type inequalities and their proofs
  7. Several inequalities of other types

Key Result

Lemma 2.1

(LiXia2017) Let $W$ be any smooth vector field on $M$. Then is a divergent form with respect to the Riemannian volume form $dv_{g}$, where $\tau=(n+1)\lambda+\mu$ and we adopt the Einstein convention.

Theorems & Definitions (30)

  • Lemma 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • proof
  • Remark 3.5
  • ...and 20 more