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 {align*} Sφ F(z) = ∫C^n F(w)eπ z· w̄φ(w-z,w̄-z) eπ |w|^2\, dw {align*} for certain φ, 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 Sφ on all polyanalytic Sobolev-Fock spaces.