Translation Invariant Operators on True Polyanalytic p-Fock Spaces

We examine translation invariant operators on the True Polyanalytic p -Fock spaces and show that they take the form aligned Sφ F(z) = ∫ Cⁿ F(w)eπ z· w̄φ (w-z,w̄-z) e-π |w|²\, dw aligned S ϕ F ( z ) = ∫ C n F ( w ) e π z · w ¯ ϕ ( w - z , w ¯ - z ) e - π | w | 2 d w for certain φ ϕ , using tools from time-frequency analysis. This extends the results of [1] to both the p -Fock spaces and the true polyanalytic p -Fock spaces. We use results on symbol classes of pseudo-differential operators to give sufficient conditions for boundedness of Sφ S ϕ on all true polyanalytic p -Fock spaces.