Integral-type operators from a mixed norm space to a Bloch-type space on the unit ball

Let $\mathbb B$ be the unit ball in $\mathbb C^n$ and let $H(\mathbb B)$ be the space of all holomorphic functions on $\mathbb B$. We introduce the following integral-type operator on $H(\mathbb B)$:
$$ I^g_\varphi(f)(z)=\int^1_0\mathrm{Re}f(\varphi(tz))g(tz)\,\frac{dt}t,\qquad z\in\mathbb B, $$
where $g\in H(\mathbb B)$, $g(0)=0$, and $\varphi$ is a holomorphic self-map of $\mathbb B$. Under study are the boundedness and compactness of the operator from the mixed norm space $H(p,q,\phi)(\mathbb B)$ to the Bloch-type space $\mathscr B_\mu(\mathbb B)$.