Sharp results for the Weyl product on modulation spaces (Q457611)

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.