Composition operators on the Privalov spaces of the unit ball of \(\mathbb{C}^n\). (English) Zbl 1062.32004

Summary: Let \(B\) and \(S\) be the unit ball and the unit sphere in \(\mathbb{C}^n\), respectively. Let \(\sigma\) be the normalized Lebesgue measure on \(S\).
Define the Privalov spaces \(N^p(B)\) \((1<p< \infty)\) by \[ N^p(B)=\left\{f\in H(B):\sup_{0 <r<1}\int_S\left\{\log \left(1+\left| f(r\zeta) \right|\right) \right\}^p d\sigma(\zeta)< \infty\right\}, \] where \(H(B)\) is the space of all holomorphic functions in \(B\). Let \(\varphi\) be a holomorphic self-map of \(B\). Let \(\mu\) denote the pull-back measure \(\sigma\circ(\varphi^*)^{-1}\).
We prove that the composition operator \(C_\varphi\) is metrically bounded on \(N^p(B)\) if and only if \(\mu({\mathcal S}(\zeta,\delta))\leq C\delta^n\) for some constant \(C\) and \(C_\varphi\) is metrically compact on \(N^p(B)\) if and only if \(\mu({\mathcal S} (\zeta,\delta))=o(\delta^n)\) as \(\delta\downarrow 0\) uniformly in \(\zeta\in S\).
Our results are an analogous results for MacCluer’s Carleson-measure criterion for the boundedness or compactness of \(C_\varphi\) on the Hardy spaces \(H^p(B)\).


32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
47B33 Linear composition operators
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