Hadamard gap series in weighted-type spaces on the unit ball

Abstract

We give a sufficient and necessary condition for an analytic function $f(z)$ on the unit ball $\BB$ in $\CC^n$ with Hadamard gaps, that is, for $f(z)={\sum }_{k=1}^{\infty }{P}_{n_k}(z)$ where $P_{n_k}(z)$ is a homogeneous polynomial of degree $n_k$ and $n_{k+1}/n_k\ge c>1$ for all $k \in \NN$, to belong to the weighted-type space $H_{\mu }^{\infty }$ and the corresponding little weighted-type space $H_{\mu ,0}^{\infty }$, under some condition posed on the weighted funtion $\mu$. We also study the growth rate of those functions in $H_{\mu }^{\infty }$. Finally, we characterize the boundedness and compactness of weighted composition operator from weighted-type space $H_{\mu }^{\infty }$ to mixed norm spaces.