Heisenberg-smooth operators from the phase space perspective

TL;DR

This paper offers a new, flexible proof of Cordes' characterization of Heisenberg-smooth operators using phase space formalism, extending the results to general quantization schemes, geometries, Schatten classes, and Heisenberg-analytic operators.

Contribution

It introduces a generalized phase space approach to characterize Heisenberg-smooth operators, broadening the scope of Cordes' original result.

Findings

New proof of Cordes' characterization using phase space formalism

Extension to general quantization schemes and phase space geometries

Derivation of Schatten class analogs and characterization of Heisenberg-analytic operators

Abstract

Cordes' characterization of Heisenberg-smooth operators bridges a gap between the theory of pseudo-differential operators and quantum harmonic analysis (QHA). We give a new proof of the result by using the phase space formalism of QHA. Our argument is flexible enough to generalize Cordes' result in several directions: (1) We can admit general quantization schemes, (2) allow for other phase space geometries, (3) obtain Schatten class analogs of the result, and (4) are able to characterize precisely 'Heisenberg-analytic' operators. For (3), we use QHA to derive Schatten versions of the Calder\'on-Vaillancourt theorem, which might be of independent interest.