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arXiv · 2410.00458

Heisenberg-smooth operators from the phase space perspective

Robert Fulsche·Lauritz van Luijk
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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.

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Robert Fulsche, Lauritz van Luijk. 2024-10-01. Heisenberg-smooth operators from the phase space perspective. https://doi.org/10.1002/mana.70019

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