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Dyadic microlocal partition for anisotropic metrics and uniform Weyl quantization

Vicente Vergara

Abstract

We develop a dyadic microlocal partition adapted to position-dependent anisotropic metrics in phase space and prove uniform bounds for localized Weyl quantization and for Moyal truncation with explicit control of remainders. A semiclassical, per-band renormalization recovers the exact scaling on each frequency band, while a Cotlar-Stein almost-orthogonality scheme ensures global convergence and operator-norm estimates with transparent dependence on finitely many symbol seminorms. The approach is robust in non-homogeneous settings where anisotropy varies with position. As applications, we construct a microlocal parametrix via truncated Moyal expansion and develop a local-global analysis of the Radon transform viewed as a model Fourier integral operator. The results provide a constructive toolset for uniform localization, composition, and recombination of pseudodifferential and Fourier integral operators.

Dyadic microlocal partition for anisotropic metrics and uniform Weyl quantization

Abstract

We develop a dyadic microlocal partition adapted to position-dependent anisotropic metrics in phase space and prove uniform bounds for localized Weyl quantization and for Moyal truncation with explicit control of remainders. A semiclassical, per-band renormalization recovers the exact scaling on each frequency band, while a Cotlar-Stein almost-orthogonality scheme ensures global convergence and operator-norm estimates with transparent dependence on finitely many symbol seminorms. The approach is robust in non-homogeneous settings where anisotropy varies with position. As applications, we construct a microlocal parametrix via truncated Moyal expansion and develop a local-global analysis of the Radon transform viewed as a model Fourier integral operator. The results provide a constructive toolset for uniform localization, composition, and recombination of pseudodifferential and Fourier integral operators.
Paper Structure (34 sections, 26 theorems, 222 equations)

This paper contains 34 sections, 26 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Anisotropic metric and smoothness of the norm
  4. Multi-indices and Leibniz’ rule.
  5. Euclidean norms and operator norm.
  6. Higher-order derivatives and tensor bounds.
  7. Microlocal conventions.
  8. Anisotropic metric and normalization.
  9. General assumptions
  10. Phase-space metric (anisotropic model)
  11. Dyadic microlocal decomposition
  12. Maximal $1/2$-separated nets in the dyadic annulus.
  13. Fibre centres via $T_x$.
  14. Mother cut and seminorms.
  15. Precuts adapted to $T_x$.
  16. ...and 19 more sections

Key Result

Lemma 2.2

Under the previous hypotheses, for every multi-index $\gamma$ there exists $C_\gamma>0$ (depending only on $\lambda^{\min}_G,\lambda^{\max}_G,|\gamma|$ and on the constants $c_\beta$ with $|\beta|\le|\gamma|$) such that

Theorems & Definitions (57)

  • Definition 2.1: Anisotropic symbol classes
  • Lemma 2.2: Derivatives of the matrix square root
  • proof
  • Proposition 2.3: Algebraic properties and comparisons
  • proof
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 47 more