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The Calderón-Zygmund inequalities on evolving Riemannian manifolds

Yongheng Han, Bing Wang

Abstract

The Calderón-Zygmund inequality is a cornerstone of harmonic analysis and partial differential equations. In this article, we establish various Calderón-Zygmund inequalities on evolving Riemannian manifolds with bounded curvature. We also provide concrete applications of such inequalities.

The Calderón-Zygmund inequalities on evolving Riemannian manifolds

Abstract

The Calderón-Zygmund inequality is a cornerstone of harmonic analysis and partial differential equations. In this article, we establish various Calderón-Zygmund inequalities on evolving Riemannian manifolds with bounded curvature. We also provide concrete applications of such inequalities.
Paper Structure (7 sections, 44 theorems, 394 equations, 4 figures)

This paper contains 7 sections, 44 theorems, 394 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Heat equations in harmonic atalas
  4. Heat kernel estimate
  5. Singular integrals and maximal functions
  6. The Calderón-Zygmund inequalities
  7. Applications

Key Result

Theorem 1.1

Suppose $\{(M,g(t)), -1 \leq t \leq 0\}$ is an evolving manifold satisfying (eqn:HA02_1). Suppose $\mathcal{T}$ is a $(C_1,C_2)$-Calderón-Zygmund integral operator. For each $p \in [2, \infty)$, there is a positive constant $C=C(n,p,\Lambda_0,C_1,C_2)$ such that for every $f \in C^{\infty}(\mathcal{M}, \mathcal{F})$. In particular, if $f$ is supported in $\mathcal{Q}'$, then

Figures (4)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:

Theorems & Definitions (79)

  • Theorem 1.1: Main result
  • Corollary 1.2
  • Corollary 1.3
  • proof : Outline of proof of Theorem \ref{['thm:HB09_1']}:
  • proof : Outline of proof of Corollary \ref{['cly:HA02_2']}:
  • proof : Outline of proof of Corollary \ref{['cly:HB09_2']}:
  • Theorem 2.1: Bishop-Gromov
  • Lemma 2.2
  • proof
  • Lemma 2.3: Vitali covering
  • ...and 69 more