Sharp results for the Weyl product on modulation spaces. (English) Zbl 1317.35307
Let \(Op^w(a)\) be a pseudodifferential operator with Weyl symbol \(a\). The Weyl product \(a_1\sharp a_2\) of the two symbols is a symbol of the product operator \(Op^w(a_1)\circ Op^w(a_2)\). The authors look for necessary and sufficient conditions such that the map \((a_1,a_2)\mapsto a_1\sharp a_2\) is well-defined and continuous. More precisely, they look for the conditions when the map extends from the product of Gelfand-Shilov spaces to the product of modulation spaces related to the symplectic Fourier transform. As a byproduct, they obtain sharp conditions for the twisted convolution to be bounded on Wiener amalgam spaces.
Reviewer: Leszek Skrzypczak (Poznań)
MSC:
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
| 42B35 | Function spaces arising in harmonic analysis |
| 44A35 | Convolution as an integral transform |
Keywords:
pseudodifferential operators; Weyl calculus; modulation spaces; symplectic Fourier transform; twisted convolution; amalgam spacesReferences:
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