Equivalent norms on Fock spaces with some application to extended Cesàro operators (Q2839367)

For \(p, \gamma>0\), the Fock space \(F^p_\gamma\) consists of those holomorphic functions \(f\) in \(\mathbb{C}^n\) for which NEWLINE\[NEWLINE \|f\|_{p, \gamma}^p = \int_{\mathbb{C}^n} \left|f(z) e^{-\frac{\gamma |z|^2}{2}} \right|^p\, dA(z) < \infty, NEWLINE\]NEWLINE where \(dA(z)\) denotes the volume measure on \(\mathbb{C}^n\). Given a positive integer \(m\), the author proves that the Fock norm \(\|\cdot\|_{p, \gamma}^p\) is equivalent to NEWLINE\[NEWLINE \sum_{|\alpha|\leq m-1} |\partial^\alpha f(0)| + \left(\sum_{|\alpha|=m} \int_{\mathbb{C}^n} \left|\partial^\alpha f(z) (1+|z|)^{-m} e^{-\frac{\gamma |z|^2}{2}} \right|^p\, dA(z)\right)^{\frac{1}{p}}. NEWLINE\]NEWLINE As an application, the author characterizes those symbols \(g\) for which the extended Cesàro operator \(T_g: F^p_\gamma \to F^q_\gamma\) is bounded or compact.