Composition operators in the Lipschitz Space of the Polydiscs

In 1987, Shapiro shew that composition operator induced by symbol $ϕ$ is compact on the Lipschltz space if and only if the infinity norm of $ϕ$ is less than 1 by a spectral-theoretic argument, where $ϕ$ is a holomorphic self-map of the unit disk. In this paper, we shall generalize Shapiro's result to the $n$-dimensional case.

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