Gaussian integral means in the Fock space (Q2063003)

For \(0<\gamma\), \(p<\infty\), the Fock space \(F_\gamma^p\) consists of all entire functions \(f\) in the complex plane \(\mathbb C\) for which \[ \|f\|_\gamma=\left(\frac {1}{v_\gamma(\mathbb C)}\int_{\mathbb C}|f(z)|^pdv_\gamma(z)\right)^{1/p}<\infty, \] where \(dv_\gamma(z)=e^{-\gamma|z|^2}dm(z)\) and \(dm\) is the Euclidean area measure on \(\mathbb C\). Let \(\mathbb D_r=\{z\in\mathbb C: |z|<r\}\). For an entire function \(f\), the Gaussian integral means of \(f\) is defined as \[ G_{p,\gamma}(r,f)=\left(\frac {1}{v_\gamma(\mathbb D_r)}\int_{\mathbb D_r}|f(z)|^pdv_\gamma(z)\right)^{1/p}. \] Given entire functions \(f\) and \(g\), respectively with Taylor coefficients \((a_n)_{n\ge 0}\) and \((b_n)_{n\ge 0}\), their Hadamard product is defined by \[ (f\ast g)(z)=\sum_{n=0}^\infty a_nb_nz^n, \quad z\in\mathbb C. \] In this paper, the authors study the relation between Hadamard products of two entire functions in the weighted Fock space \(F_\gamma^p\) and their integral Gaussian means. Their main result shows that the growth of the Gaussian means of the Hadamard product \(f\ast g\ast k_{\gamma,r}\) is controlled by the growth of Gaussian means of \(f\) and \(g\); that is, if \(f\in F_\gamma^p\) and \(g\in F_\gamma^q\) with \(0<\gamma<\infty\) and \(1\le p\le q<\infty\), then \[ G_{q,\gamma}(s, f\ast g\ast k_{\gamma,r})\le G_{p,\gamma}(r, f)G_{q,\gamma}(s, g) \] for all \(r,s\in (0,\infty)\), where \(k_{\gamma,r}\) is a key auxiliary function.