Weighted theory of Toeplitz operators on the Fock spaces
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912991095554048 |
|---|---|
| author | Chen, Jiale |
| author_facet | Chen, Jiale |
| contents | We study the weighted compactness and boundedness of Toeplitz operators on the Fock spaces. Fix $α>0$. Let $T_φ$ be the Toeplitz operator on the Fock space $F^2_α$ over $\mathbb{C}^n$ with symbol $φ\in L^{\infty}$. For $1<p<\infty$ and any finite sum $T$ of finite products of Toeplitz operators $T_φ$'s, we show that $T$ is compact on the weighted Fock space $F^p_{α,w}$ if and only if its Berezin transform vanishes at infinity, where $w$ is a restricted $A_p$-weight on $\mathbb{C}^n$. Concerning boundedness, for $1\leq p<\infty$, we characterize the $r$-doubling weights $w$ such that $T_φ$ is bounded on the weighted spaces $L^p_{α,w}$ via a $φ$-adapted $A_p$-type condition. Our method also establishes a two weight inequality for the Fock projections in the case of $r$-doubling weights. Moreover, we characterize the corresponding weighted compactness of Bergman--Toeplitz operators, which answers a question raised by Stockdale and Wagner [Math. Z. 305 (2023), no. 1, Paper No. 10]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13571 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weighted theory of Toeplitz operators on the Fock spaces Chen, Jiale Functional Analysis We study the weighted compactness and boundedness of Toeplitz operators on the Fock spaces. Fix $α>0$. Let $T_φ$ be the Toeplitz operator on the Fock space $F^2_α$ over $\mathbb{C}^n$ with symbol $φ\in L^{\infty}$. For $1<p<\infty$ and any finite sum $T$ of finite products of Toeplitz operators $T_φ$'s, we show that $T$ is compact on the weighted Fock space $F^p_{α,w}$ if and only if its Berezin transform vanishes at infinity, where $w$ is a restricted $A_p$-weight on $\mathbb{C}^n$. Concerning boundedness, for $1\leq p<\infty$, we characterize the $r$-doubling weights $w$ such that $T_φ$ is bounded on the weighted spaces $L^p_{α,w}$ via a $φ$-adapted $A_p$-type condition. Our method also establishes a two weight inequality for the Fock projections in the case of $r$-doubling weights. Moreover, we characterize the corresponding weighted compactness of Bergman--Toeplitz operators, which answers a question raised by Stockdale and Wagner [Math. Z. 305 (2023), no. 1, Paper No. 10]. |
| title | Weighted theory of Toeplitz operators on the Fock spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2501.13571 |