A class of Berezin-type operators on weighted Fock spaces with $A_{\infty}$-type weights
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913488526376960 |
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| author | Chen, Jiale |
| author_facet | Chen, Jiale |
| contents | Let $0<α,β,t<\infty$ and $μ$ be a positive Borel measure on $\mathbb{C}^n$. We consider the Berezin-type operator $S^{t,α,β}_μ$ defined by $$S^{t,α,β}_μf(z):=\left(\int_{\mathbb{C}^n}e^{-\fracβ{2}|z-u|^2}|f(u)|^te^{-\frac{αt}{2}|u|^2}dμ(u)\right)^{1/t},\quad z\in\mathbb{C}^n.$$ We completely characterize the boundedness and compactness of $S^{t,α,β}_μ$ from the weighted Fock space $F^p_{α,w}$ into the Lebesgue space $L^q(wdv)$ for all possible indices, where $w$ is a weight on $\mathbb{C}^n$ that satisfies an $A_{\infty}$-type condition. This solves an open problem raised by Zhou, Zhao and Tang [Banach J. Math. Anal. 18 (2024), Paper No. 20]. As an application, we obtain the description of the boundedness and compactness of Toeplitz-type operators acting between weighted Fock spaces induced by $A_{\infty}$-type weights. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_01132 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A class of Berezin-type operators on weighted Fock spaces with $A_{\infty}$-type weights Chen, Jiale Functional Analysis Let $0<α,β,t<\infty$ and $μ$ be a positive Borel measure on $\mathbb{C}^n$. We consider the Berezin-type operator $S^{t,α,β}_μ$ defined by $$S^{t,α,β}_μf(z):=\left(\int_{\mathbb{C}^n}e^{-\fracβ{2}|z-u|^2}|f(u)|^te^{-\frac{αt}{2}|u|^2}dμ(u)\right)^{1/t},\quad z\in\mathbb{C}^n.$$ We completely characterize the boundedness and compactness of $S^{t,α,β}_μ$ from the weighted Fock space $F^p_{α,w}$ into the Lebesgue space $L^q(wdv)$ for all possible indices, where $w$ is a weight on $\mathbb{C}^n$ that satisfies an $A_{\infty}$-type condition. This solves an open problem raised by Zhou, Zhao and Tang [Banach J. Math. Anal. 18 (2024), Paper No. 20]. As an application, we obtain the description of the boundedness and compactness of Toeplitz-type operators acting between weighted Fock spaces induced by $A_{\infty}$-type weights. |
| title | A class of Berezin-type operators on weighted Fock spaces with $A_{\infty}$-type weights |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2409.01132 |