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Matrix representation of Magnetic pseudo-differential operators via tight Gabor frames

Horia D. Cornean, Bernard Helffer, Radu Purice

Abstract

In this paper we use some ideas from \cite{FG-97, G-06} and consider the description of Hörmander type pseudo-differential operators on $\mathbb{R}^d$ ($d\geq1$), including the case of the magnetic pseudo-differential operators introduced in \cite{IMP-1, IMP-19}, with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calder{ó}n-Vaillancourt theorem and Beals' commutator criterion, and also establish local trace-class criteria.

Matrix representation of Magnetic pseudo-differential operators via tight Gabor frames

Abstract

In this paper we use some ideas from \cite{FG-97, G-06} and consider the description of Hörmander type pseudo-differential operators on (), including the case of the magnetic pseudo-differential operators introduced in \cite{IMP-1, IMP-19}, with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calder{ó}n-Vaillancourt theorem and Beals' commutator criterion, and also establish local trace-class criteria.
Paper Structure (14 sections, 9 theorems, 87 equations)

This paper contains 14 sections, 9 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Main goals
  3. General notation
  4. The magnetic field.
  5. The Gabor frame
  6. Definition
  7. Properties of the magnetic Gabor frame.
  8. Infinite matrices associated with operators in a magnetic Gabor frame.
  9. The matrix form of the magnetic Weyl calculus in a magnetic Gabor frame.
  10. Brief reminder of the magnetic Weyl calculus.
  11. The main results
  12. A magnetic version of the Calderón-Vaillancourt Theorem.
  13. On the Beals commutator criterion
  14. Local Schatten-class properties

Key Result

Proposition 2.2

We have the following properties: (i). The Gabor frame $\{\mathcal{G}^A_{\tilde{\gamma}} \}$ indexed by $\tilde{\gamma}:=(\gamma,\gamma^*)\in\widetilde{\Gamma}$ is a Parseval frame in $L^2(\mathcal{X})$, i.e. the map: is an isometry. (ii). Given the magnetic Gabor frame $\{\mathcal{G}^A_{\tilde{\gamma}} \}$, we have for any $f\in L^2(\mathcal{X})$ the identity: with the above series converging

Theorems & Definitions (18)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.4
  • proof
  • Theorem 3.1
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 8 more