Equivalent norms on Fock spaces with some application to extended Cesaro operators

Abstract:

Let $F_\gamma ^p$ be the Fock space of all holomorphic functions $f$ in ${\mathbf C}^n$ with the Fock norm \[ \|f\|_{p, \gamma }=\left \{\int _{\mathbf {C}^n}\left |f(z)e^ {-\frac {\gamma |z|^2}{2}}\right |^{p}dA(z)\right \}^{\frac {1}{p}} <\infty , \] where $p, \gamma$ are positive numbers. We prove that, given any positive integer $m$, the Fock norm $\|f\|_{p, \gamma }$ is equivalent to \[ \sum _{|\alpha |\le m-1} |\partial ^\alpha f(0)|+ \left \{\sum _{|\alpha |= m} \int _{{\mathbf C}^n } \left | \partial ^\alpha f(z) (1+|z|)^{-m} e^ {-\frac {\gamma |z|^2}{2}} \right |^p dA(z)\right \}^{\frac 1p}.\] As some application we characterize these holomorphic functions $g$ in ${\mathbf C}^n$ for which the induced extended Cesaro operator $T_g$ is bounded (or compact) from one Fock space $F_\gamma ^p$ to another $F_\gamma ^q$.