We examine translation invariant operators on the Polyanalytic Sobolev-Fock spaces and show that they take the form \begin{ align*} S_{ \phi} F(z) = \int_{ \mathbb{ C}^n} F(w)e^{ \pi z\cdot \overline{ w}}\phi(w-z,\overline{ w}-z) e^{ \pi |w|^2}\, dw \end{ align*} for certain $\phi$, using tools from time-frequency analysis. This extends the results of Cao et al. (2020) to both the Sobolev-Fock spaces and the Polyanalytic Sobolev-Fock spaces. We use results on symbol classes of pseudo-differential operators to give sufficient conditions for boundedness of $S_{ \phi}$ on all polyanalytic Sobolev-Fock spaces.
Categories: math.FA math.CV