Bergman type operators on some generalized Cartan-Hartogs domains
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For $\mu=(\mu_1,\dots,\mu_t)$ ($\mu_j>0$), $\xi=(z_1,\dots,z_t,w)\in \mathbb{C}^{n_1}\times\cdots\times \mathbb{C}^{n_t}\times \mathbb{C}^m$, define $$\Omega(\mu,t)\!=\!\big\{\xi\in\mathbb{B}_{n_1}\times\cdots\times\mathbb{B}_{n_t}\times\mathbb{C}^{m}: \|w\|^2<C(\chi,\mu)\prod\nolimits_{j=1}^{t}(1-\|z_j\|^2)^{\mu_j}\big\},$$ where $\mathbb{B}_{n_j}$ is the unit ball in $\mathbb{C}^{n_j}$ $(1\le j\le t)$, $C(\chi,\mu)$ is a constant only depending on $\chi=(n_1,\dots,n_t)$ and $\mu=(\mu_1,\dots,\mu_t)$, which is a special type of generalized Cartan-Hartogs domain. We will give some sufficient and necessary conditions for the boundedness of some type of operators on $L^p(\Omega(\mu,t), \omega)$ (the weighted $L^p$ space of $\Omega(\mu,t)$ with weight $\omega$, $1<p<\infty$). This result generalizes the works from certain classes of generalized complex ellipsoids to the generalized Cartan-Hartogs domain $\Omega(\mu,t)$.
Abstract
For $\mu=(\mu_1,\dots,\mu_t)$ ($\mu_j>0$), $\xi=(z_1,\dots,z_t,w)\in \mathbb{C}^{n_1}\times\cdots\times \mathbb{C}^{n_t}\times \mathbb{C}^m$, define $$\Omega(\mu,t)\!=\!\big\{\xi\in\mathbb{B}_{n_1}\times\cdots\times\mathbb{B}_{n_t}\times\mathbb{C}^{m}: \|w\|^2<C(\chi,\mu)\prod\nolimits_{j=1}^{t}(1-\|z_j\|^2)^{\mu_j}\big\},$$ where $\mathbb{B}_{n_j}$ is the unit ball in $\mathbb{C}^{n_j}$ $(1\le j\le t)$, $C(\chi,\mu)$ is a constant only depending on $\chi=(n_1,\dots,n_t)$ and $\mu=(\mu_1,\dots,\mu_t)$, which is a special type of generalized Cartan-Hartogs domain. We will give some sufficient and necessary conditions for the boundedness of some type of operators on $L^p(\Omega(\mu,t), \omega)$ (the weighted $L^p$ space of $\Omega(\mu,t)$ with weight $\omega$, $1<p<\infty$). This result generalizes the works from certain classes of generalized complex ellipsoids to the generalized Cartan-Hartogs domain $\Omega(\mu,t)$.
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