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$L_p$ estimates in the Androulidakis-Mohsen-Yuncken calculus

Edward McDonald

Abstract

We prove that order zero operators in the pseudodifferential calculus associated to a filtration defined by Androulidakis, Mohsen and Yuncken are bounded on $L_p$ spaces for $1<p<\infty.$

$L_p$ estimates in the Androulidakis-Mohsen-Yuncken calculus

Abstract

We prove that order zero operators in the pseudodifferential calculus associated to a filtration defined by Androulidakis, Mohsen and Yuncken are bounded on spaces for

Paper Structure

This paper contains 14 sections, 31 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Main results
  4. Abstract kernel estimates
  5. Abstract Operator estimates
  6. Kernels derived from graded bases
  7. Metric associated to a family of vector fields
  8. Graded basis
  9. Level sets of submersions
  10. The proof of \ref{['kernelsmoothnessballvolume']} for $\boldsymbol{K}$
  11. The Androulidakis-Mohsen-Yuncken calculus
  12. Compactness of negative order operators
  13. Sobolev spaces
  14. Applications to elliptic regularity in $L_p$

Key Result

Theorem 2.1

Let $1<p<\infty,$ and let $T \in \Psi^m_{\mathcal{F}}(X).$ If $\Re(m)\leq 0,$ then $T$ is locally bounded on $L_p(X).$ Moreover, the associated family $\{T_{\hbar}\}_{\hbar>0}$ is uniformly continuous as $\hbar\to 0$ in the sense that if $\phi,\psi\in C^\infty_c(X),$ then If $\Re(m)<0,$ then the family $\{T_{\hbar}\}_{\hbar>0}$ is uniformly continuous as $\hbar\to 0$ on $L_p(X)$ for any $1\leq p\

Theorems & Definitions (72)

  • Theorem 2.1
  • Remark 2.2
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 4.1
  • ...and 62 more