Heisenberg‐smooth operators from the phase‐space perspective

Robert Fulsche and Lauritz van Luijk

Mathematische Nachrichten, 2025, vol. 298, issue 8, 2845-2866

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 Calderón–Vaillancourt theorem, which might be of independent interest.

Date: 2025
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