Translation Invariant Operators on True Polyanalytic p-Fock Spaces

Abstract We examine translation invariant operators on the True Polyanalytic p -Fock spaces and show that they take the form $$\begin{aligned} S_{\phi } F(z) = \int _{\mathbb {C}^n} F(w)e^{\pi z\cdot \overline{w}}\phi (w-z,\overline{w}-z) e^{-\pi |w|^2}\, dw \end{aligned}$$ 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 $$\phi $$ 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 $$S_{\phi }$$ S ϕ on all true polyanalytic p -Fock spaces.