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arXiv · 2407.02982

Translation Invariant Operators on Polyanalytic Sobolev-Fock Spaces

Henry McNulty
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Abstract

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 $ϕ$, 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 $S_ϕ$ on all polyanalytic Sobolev-Fock spaces.

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Henry McNulty. 2024-07-03. Translation Invariant Operators on Polyanalytic Sobolev-Fock Spaces. https://arxiv.org/abs/2407.02982

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