Translation Invariant Operators on Polyanalytic Sobolev-Fock Spaces
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911943132971008 |
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| author | McNulty, Henry |
| author_facet | McNulty, Henry |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02982 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Translation Invariant Operators on Polyanalytic Sobolev-Fock Spaces McNulty, Henry Functional Analysis Complex Variables 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. |
| title | Translation Invariant Operators on Polyanalytic Sobolev-Fock Spaces |
| topic | Functional Analysis Complex Variables |
| url | https://arxiv.org/abs/2407.02982 |