Weighted composition operators between Bloch-type spaces in the polydisc

For $p>0$, let $\mathscr B^p(\mathbb D^n)$ denote the $p$-Bloch space on the unit polydisc $\mathbb D^n$ of $\mathbb C^n$ and $\varphi(z)=(\varphi_1(z),\dots,\varphi_n(z))$ a holomorphic self-map of $\mathbb D^n$. We investigate the boundedness and compactness of the weighted composition $uC_\varphi f(z)=u(z)f(\varphi(z))$ between $p$-Bloch space $\mathscr B^p(\mathbb D^n)$ (little $p$-Bloch space $\mathscr B^p_0(\mathbb D^n)$) and $q$-Bloch space $\mathscr B^q(\mathbb D^n)$ (little $q$-Bloch space $\mathscr B^q_0(\mathbb D^n)$). The most important result in the paper is that conditions for the compactness are different for the cases $p\in(0,1)$ and $p\geqslant1$, unlike for the case of the weighted operators on the unit disc.
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