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Sheafs of ultradifferentiable functions

Stefan Fürdös

Abstract

An abstract theory of ultradifferentiable sheafs is developed. Moreover, various applications to the theory of linear partial differential equations, differential geometry and, in particular, CR geometry are discussed.

Sheafs of ultradifferentiable functions

Abstract

An abstract theory of ultradifferentiable sheafs is developed. Moreover, various applications to the theory of linear partial differential equations, differential geometry and, in particular, CR geometry are discussed.
Paper Structure (13 sections, 41 theorems, 75 equations)

This paper contains 13 sections, 41 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Sheafs of smooth functions in PDE theory
  3. Isotropic sheafs
  4. Microlocal sheafs
  5. Ultradifferentiable hypoellipticity of analytic partial differential operators
  6. Geometric sheafs of smooth functions
  7. Regular sheafs
  8. Differential geometry in the ultradifferentiable category
  9. Uniqueness theorems for quasianalytic operators
  10. Applications to CR theory
  11. Ultradifferentiable CR manifolds
  12. Regularity of Sections of CR bundles
  13. Examples of ultradifferential sheafs

Key Result

Lemma 2.3

Let $\ifstrempty{}{ \ifstrempty{}{ \mathcal{R}}{\mathcal{R}_{}} }{ \ifstrempty{}{\mathcal{R}()}{\mathcal{R}_{}()} }$ be an isotropic ultradifferentiable class. For each $n\in\mathbb{N}$ the following statements are equivalent:

Theorems & Definitions (90)

  • Definition 2.1
  • proof
  • Definition 2.2: Quasianalyticity
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 80 more