Journal of the
Korean Mathematical Society JKMS

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 \[ N^p(B) = \biggl\{\, f \in H(B) \, : \, \sup_{0 < r < 1}\int_{S}\{\log(1+|f(r\zeta)|)\}^pd\sigma(\zeta) < \infty \, \biggr\}, \] 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}^{\ast})^{-1} $. In this paper, we prove that the composition operator $ C_{\varphi} $ is $metrically$ $bounded$ on $ N^p(B) $ if and only if $ \mu(\mathcal{S}(\zeta, \delta)) \le 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) $.

Keywords: Hardy spaces, Privalov spaces, composition operators, unit ball of $ \mathbb{C}^n $

MSC numbers: Primary 32A35, 32A37; Secondary 47B33