Sum of some product-type operators from mixed-norm spaces to weighted-type spaces on the unit ball

Let $u_j$ be the holomorphic functions on the open unit ball $\mathbb{B}$ in ${\mathbb{C}}^n$, $j=\overline{0,m}$, $\phi$ a holomorphic self-map of $\mathbb{B}$, and ${\mathrm{\Re }}^j$ the $j$th iterated radial derivative operator. In this paper, the boundedness and compactness of the sum operator ${\mathfrak{S}}_{\overrightarrow{u},\phi }^m=\sum _{j=0}^m{M}_{u_j}{C}_{\phi }{\mathrm{\Re }}^j$ from the mixed-norm space $H(p,q,\varphi )$, where $0<p,q<+\mathrm{\infty }$, and $\varphi$ is normal, to the weighted-type space $H_{\mu }^{\mathrm{\infty }}$ are characterized. For the mixed-norm space $H(p,q,\varphi )$, $1\le p<+\mathrm{\infty }$, $1<q<+\mathrm{\infty }$, the essential norm estimate of the operator is given, and the Hilbert-Schmidt norm of the operator on the weighted Bergman space $A_{\alpha }^2$ is also calculated.