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Pseudodifferential operators with formal Gevrey symbols and symbolic calculus

Haoren Xiong

Abstract

We construct the parametrix of an elliptic Gevrey pseudodifferential operator, by introducing a family of norms for formal Gevrey symbols with the property of a Banach algebra under the symbol calculus. As an application, we obtain estimates for adiabatic projectors in the Gevrey setting.

Pseudodifferential operators with formal Gevrey symbols and symbolic calculus

Abstract

We construct the parametrix of an elliptic Gevrey pseudodifferential operator, by introducing a family of norms for formal Gevrey symbols with the property of a Banach algebra under the symbol calculus. As an application, we obtain estimates for adiabatic projectors in the Gevrey setting.
Paper Structure (3 sections, 6 theorems, 60 equations)

This paper contains 3 sections, 6 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Formal Gevrey symbols and pseudonorms
  3. Adiabatic projectors

Key Result

Theorem 1

Let $s,\,\sigma\geq 1$, and let $\Omega\subset{\mathbb R}^{2n}$ be open. Let $p=\sum_{k=0}^{\infty} h^k p_k$ be a formal ${\mathcal{G}}^{s,\sigma}$ symbol on $\Omega$. Suppose that $p$ is elliptic in the sense that $p_0\neq 0$ everywhere on $\Omega$. Then there is a unique formal ${\mathcal{G}}^{s,\

Theorems & Definitions (14)

  • Definition 1.1
  • Theorem 1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 4 more