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

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