Translation Invariant Operators on Polyanalytic Sobolev-Fock Spaces

We examine translation invariant operators on the Polyanalytic Sobolev-Fock spaces and show that they take the form \begin{align*} S_φ F(z) = \int_{$\mathbb{C}$^n} F(w)e^{πz\cdot \overline{w}}φ(w-z,\overline{w}-z) e^{π|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.