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Hypoellipticity of analytic differential operators in general ultradifferentiable classes

Stefan Fürdös

TL;DR

This work develops a unified ultradifferentiable microlocal framework for analytic differential operators, using weight matrices and semiregular classes to study hypoellipticity across broad ultradifferentiable categories. It generalizes Hörmander’s wavefront-transform theory and extends Metivier–Okaji-type results to these classes, proving that analytic pseudodifferential and Fourier integral operators are microlocal with respect to $\mathrm{WF}_{[\mathfrak{M}]}$ and that elliptic analytic operators are hypoelliptic in all semiregular classes. A central achievement is Treves’ characterization of hypoellipticity for principal-type operators carried over to the ultradifferentiable setting, along with Metivier-type results for operators with multiple characteristics. The results provide a broad, flexible framework for ultradifferentiable regularity, enabling finer distinctions than Gevrey or Denjoy–Carleman approaches and facilitating a parametrix-based analysis within weight-matrix classes. The methods merge microlocal analysis, distribution theory, and analytic parametrices to illuminate regularity in a wide spectrum of ultradifferentiable environments.

Abstract

We show that analytic pseudodifferential and Fourier integral operators behave well for ultradifferentiable classes satisfying minimal regularity properties. As an application we investigate the ultradifferentiable regularity properties of several examples of analytic differential operators. In particular we extend Treves' characterization of the hypoellipticity of analytic operators of principal type to the ultradifferentiable category.

Hypoellipticity of analytic differential operators in general ultradifferentiable classes

TL;DR

This work develops a unified ultradifferentiable microlocal framework for analytic differential operators, using weight matrices and semiregular classes to study hypoellipticity across broad ultradifferentiable categories. It generalizes Hörmander’s wavefront-transform theory and extends Metivier–Okaji-type results to these classes, proving that analytic pseudodifferential and Fourier integral operators are microlocal with respect to and that elliptic analytic operators are hypoelliptic in all semiregular classes. A central achievement is Treves’ characterization of hypoellipticity for principal-type operators carried over to the ultradifferentiable setting, along with Metivier-type results for operators with multiple characteristics. The results provide a broad, flexible framework for ultradifferentiable regularity, enabling finer distinctions than Gevrey or Denjoy–Carleman approaches and facilitating a parametrix-based analysis within weight-matrix classes. The methods merge microlocal analysis, distribution theory, and analytic parametrices to illuminate regularity in a wide spectrum of ultradifferentiable environments.

Abstract

We show that analytic pseudodifferential and Fourier integral operators behave well for ultradifferentiable classes satisfying minimal regularity properties. As an application we investigate the ultradifferentiable regularity properties of several examples of analytic differential operators. In particular we extend Treves' characterization of the hypoellipticity of analytic operators of principal type to the ultradifferentiable category.

Paper Structure

This paper contains 7 sections, 38 theorems, 177 equations.

Table of Contents

  1. Statement of main results
  2. Microlocal analysis in general ultradifferentiable classes
  3. Analytic pseudodifferential operators and ultradifferentiable microdistributions
  4. Hypoelliptic differential operators of principal type
  5. Metivier Operators
  6. Differential operators with constant coefficients
  7. A semiregular quasianalytic class transversal to all semiregular Denjoy-Carleman classes

Key Result

Theorem 1

Let $P$ be an analytic differential operator of principal type and $\mathcal{U}$ be a semiregular ultradifferentiable class. Then $P$ is smooth hypoelliptic if and only if $P$ is hypoelliptic with respect to $\mathcal{U}$.

Theorems & Definitions (68)

  • Theorem
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • ...and 58 more