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COMPOSITION OPERATORS ON THE PRIVALOV SPACES OF THE UNIT BALL OF ℂ<sup>n</sup>
UEKI SEI-ICHIRO Korean Mathematical Society 2005 대한수학회지 Vol.42 No.1
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 $$N^P(B)\;=\;\{\;f\;{\in}\;H(B) : \sup_{0<r<1}{\Bigint}_S \{log(1+|f(\gamma\zeta)|)\}^{p}d\sigma(\zeta)\;<\;\infty\;\}$$ 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 ${\sigma}o({\varphi}^{\ast})^{-1}$. In this paper, we prove that the composition operator $C_{\varphi}$ is metrically bounded on $N^P$(B) if and only if ${\mu}(S(\zeta,\delta)){\le}C{\delta}^n$ for some constant C and $C_{\varphi}$ is metrically compact on $N^P(B)$ if and only if ${\mu}(S(\zeta,\delta))=o({\delta}^n)$ as ${\delta}\;{\downarrow}\;0$ uniformly in ${\zeta}\;\in\;S. Our results are an analogous results for Mac Cluer's Carleson-measure criterion for the boundedness or compactness of $C_{\varphi}$ on the Hardy spaces $H^P(B)$.
Hideto Ueki,Takuto Hara,Yasuyoshi Okamura,Yukari Bando,Tomoaki Terakawa,Junya Furukawa,Kenichi Harada,Yuzo Nakano,Masato Fujisawa 대한비뇨의학회 2022 Investigative and Clinical Urology Vol.63 No.4
Purpose: Two methods are used to identify sarcopenia by calculating skeletal muscle area on computed tomography: the skeletal muscle index (SMI) and the psoas muscle index (PMI). Programmed death (PD)-1 inhibitors are helpful in treating metastatic renal cell carcinoma (mRCC). However, there remains insufficient information regarding a clear and easy-to-use biomarker for predicting the response to PD-1 inhibitors in patients with mRCC. Therefore, we investigated the influence of sarcopenia on clinical outcomes in patients with mRCC undergoing treatment with nivolumab. Materials and Methods: This study evaluated 96 patients with RCC who received nivolumab. The SMI and PMI were calculated for each patient and normalized for stature by use of the following formulas: SMI (cm2/m2)=([skeletal muscle cross-sectional area at the level of L3]/[height]2) and PMI (cm2/m2) = ([left-right sum of the psoas muscle areas at the level of L3]/[height]2). The relationship of the clinical variables with progression-free survival and overall survival (OS) was examined using a Cox proportional hazards model. Results: According to the SMI-based definition of sarcopenia, 74.0% of patients had sarcopenia. However, according to the PMI-based definition of sarcopenia, only 34.3% of patients were diagnosed with sarcopenia. Multivariate analysis identified sarcopenia based on PMI (hazard ratio [HR], 3.85; 95% confidence interval [CI], 2.04–7.26; p<0.001) and International Metastatic RCC Database Consortium poor risk status (HR, 1.90; 95% CI, 1.03–3.50; p=0.041) as significant and independent prognostic factors of OS. Conclusions: PMI-based sarcopenia is a significant prognostic factor for OS in patients with RCC who receive nivolumab therapy.