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 $H^\infty_\mu$ and the corresponding little weighted-type space $H^\infty_{\mu, 0}$, under some condition posed on the weighted funtion $\mu$. We also study the growth rate of those functions in $H^\infty_\mu$. Finally, we characterize the boundedness and compactness of weighted composition operator from weighted-type space $H^\infty_\mu$ to mixed norm spaces.