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Almost diagonalization of $Ψ$DO's over various generalized function spaces

Stevan Pilipović, Nenad Teofanov, Filip Tomić

Abstract

Inductive and projective type sequence spaces of sub- and super-exponential growth, and the corresponding inductive and projective limits of modulation spaces are considered as a framework for almost diagonalization of pseudo-differential operators. Moreover, recent results of the first author and B. Prangoski related to the almost diagonalization of pseudo-differential operators in the context of Hörmander metrics are reviewed.

Almost diagonalization of $Ψ$DO's over various generalized function spaces

Abstract

Inductive and projective type sequence spaces of sub- and super-exponential growth, and the corresponding inductive and projective limits of modulation spaces are considered as a framework for almost diagonalization of pseudo-differential operators. Moreover, recent results of the first author and B. Prangoski related to the almost diagonalization of pseudo-differential operators in the context of Hörmander metrics are reviewed.
Paper Structure (14 sections, 8 theorems, 74 equations)

This paper contains 14 sections, 8 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Preliminaries
  4. Notation
  5. Weights
  6. STFT and modulation spaces
  7. Pseudo-differential operators
  8. Novel spaces for symbols of pseudo-differential operators
  9. Spaces of sequences
  10. Wiener amalgam spaces related to $\mathcal{A}_r^s$
  11. Modulation spaces related to $\mathcal{A}_r^s$
  12. Approximate diagonalization in Gelfand-Shilov spaces
  13. Extensions on the spaces with the Hörmander metric
  14. The symplectic short-time Fourier transform

Key Result

Lemma 2.1

If $m=m_{a, b, c, t}$ is given by PrimerTezina then

Theorems & Definitions (12)

  • Example 2.1
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.1
  • Theorem 4.1
  • Theorem 4.2
  • ...and 2 more