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Let $D$ be a bounded homogeneous domain in $\mathbb{C}^n$. In this paper, westudy the bounded and the compact weighted composition operators mapping theHardy space $H^\infty(D)$ into the Bloch space of $D$. We characterize thebounded weighted composition operators, provide operator norm estimates, andgive sufficient conditions for compactness. We prove that these conditions arenecessary in the case of the unit ball and the polydisk. We then show that if$D$ is a bounded symmetric domain, the bounded multiplication operators from$H^\infty(D)$ to the Bloch space of $D$ are the operators whose symbol isbounded.

Weighted composition operators from $H^\infty$ to the Bloch space of a bounded homogeneous domain