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 ${\mathbb{R}}^d$ ($d\ge 1$), 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.

arXiv comment17 pages, accepted for publication in Journal of Fourier Analysis and Applications

References in corpus (7)

• The Magnetic Weyl Calculus
• Magnetic Pseudodifferential Operators
• Parseval frames of exponentially localized magnetic Wannier functions
• On the construction of Wannier functions in topological insulators: the 3D case
• Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices
• A Beals criterion for magnetic pseudodifferential operators proved with magnetic Gabor frames
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Cited by in corpus (1)

• Magnetic Pseudo-differential Operators with Hörmander Symbols Dominated by Tempered Weights