Volume integral means of holomorphic functions

Abstract:

The classical integral means of a holomorphic function $f$ in the unit disk are defined by \[ \Bigg [\frac 1{2\pi }\int _0^{2\pi }|f(re^{i\theta })|^p d\theta \Bigg ]^{1/p}, \quad 0\le r<1.\] These integral means play an important role in modern complex analysis. In this note we consider integral means of holomorphic functions in the unit ball $\mathbb B_n$ in $\mathbb C^n$ with respect to weighted volume measures, \[ M_{p,\alpha }(f,r)=\left [\frac {1}{v_\alpha (r\mathbb B_n)} \int _{r\mathbb B_n}|f(z)|^p dv_\alpha (z)\right ]^{1/p}, \quad 0\le r<1,\] where $\alpha$ is real, $dv_\alpha (z)=(1-|z|^2)^\alpha dv(z)$, and $dv$ is volume measure on $\mathbb B_n$. We show that $M_{p,\alpha }(f,r)$ increases with $r$ strictly unless $f$ is a constant, but in contrast with the classical case, $\log M_{p,\alpha }(f,r)$ is not always convex in $\log r$. As an application, we show that if $\alpha \le -1$, $M_{p,\alpha }(f,r)$ is bounded in $r$ if and only if $f$ belongs to the Hardy space $H^p$, while if $\alpha >-1$, $M_{p,\alpha }(f,r)$ is bounded in $r$ if and only if $f$ is in the weighted Bergman space $A^p_\alpha$.