Higher order derivative characterization for Fock-type spaces (Q907869)

For \(\alpha>0\) and \(n\in\mathbb{N}\), the Fock-type space \(\mathcal{F}_\alpha^\infty(\mathbb{C}^n)\) consists of those entire functions \(f\) on \(\mathbb{C}^n\) for which \[ \|f\|_{\infty, \alpha} =\sup_{z\in\mathbb{C}^n} |f(z)|\exp\left(-\frac{\alpha}{2} |z|^2 \right) < \infty. \] The author characterizes \(\mathcal{F}_\alpha^\infty(\mathbb{C}^n)\) in terms of higher-order derivatives of \(f\). As an application, he studies the boundedness and compactness of the extended Cesàro operator on \(\mathcal{F}_\alpha^\infty(\mathbb{C}^n)\) and \(\mathcal{F}_{\alpha, 0}^\infty(\mathbb{C}^n)\).