Heisenberg-smooth operators from the phase-space perspective

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.