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The Covariant Riesz Transforms on Riemannian Manifolds

Yongheng Han, Bing Wang

Abstract

We establish the $L^p$-boundedness of the local covariant Riesz transform for differential forms on manifold $M$ with bounded $\|Rm\|$. Let $Δ_j$ be the Hodge Laplace operator on $j$-forms. For any $p \in (1, \infty)$ and $κ>κ_0$, we show that the operator $\nabla (Δ_j + κ)^{-1/2}$ is bounded on $L^p(M)$. Consequently, we obtain Calderón-Zygmund estimates for manifolds with bounded Riemannian curvature.

The Covariant Riesz Transforms on Riemannian Manifolds

Abstract

We establish the -boundedness of the local covariant Riesz transform for differential forms on manifold with bounded . Let be the Hodge Laplace operator on -forms. For any and , we show that the operator is bounded on . Consequently, we obtain Calderón-Zygmund estimates for manifolds with bounded Riemannian curvature.
Paper Structure (10 sections, 28 theorems, 180 equations, 1 figure)

This paper contains 10 sections, 28 theorems, 180 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Volume comparison theorem
  4. Hardy–Littlewood maximal operators
  5. Hodge Laplacian and rough Laplacian
  6. Harmonic radius
  7. Heat equations in $W^{2,1}_p$ atalas
  8. Heat kernel estimate
  9. Boundedness of Riesz type operator
  10. Proof of Theorem \ref{['thm1.4']}

Key Result

Theorem 1.1

Bak85 Assume that $\|Rm\|\leq \Lambda_0$, then there exists a $\kappa_0=\kappa_0(\Lambda_0,n)>0$ such that for all $\kappa\geq \kappa_0$ and $j=\{0,\cdots,n\}$ the operators $d_j(\Delta_j+\kappa )^{-1/2}$ and $d_{j-1}^*(\Delta_j +\kappa)^{-1/2}$ are weak $(1,1)$. For every $p\in (1,+\infty)$, one ha with norm bounds depending on $n,p,\Lambda_0, \kappa$. Here, $d_j$ denotes the exterior derivative

Figures (1)

  • Figure 1:

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Main result
  • Remark 1.4
  • Theorem 1.5: Calderón-Zygmund inequality
  • Remark 1.6
  • Theorem 1.7
  • Theorem 2.1
  • Corollary 2.2
  • proof
  • ...and 41 more