Let Un be the unit polydisc of ?n and φ=(φ1, ?, φ n ) a holomorphic self-map of Un. As the main result of the paper, it shows that the composition operator C is compact on the Bloch space β(Un) if and only if for ...Let Un be the unit polydisc of ?n and φ=(φ1, ?, φ n ) a holomorphic self-map of Un. As the main result of the paper, it shows that the composition operator C is compact on the Bloch space β(Un) if and only if for every ε > 0, there exists a δ > 0, such that $$\sum\limits_{k,1 = 1}^n {\left| {\frac{{\partial \phi _l }}{{\partial z_k }}(z)} \right|} \frac{{1 - |z_k |^2 }}{{1 - |\phi _l (z)|^2 }}< \varepsilon ,$$ whenever dist(φ(z), ?U n )<δ.展开更多
Let Un be the unit polydisc of Cn and φ = (φ 1,...,φ n ) a holomorphic self-map of Un. Let 0 ≤α 1. This paper shows that the composition operator C is bounded on the Lipschitz space Lip(Un) if and only if there e...Let Un be the unit polydisc of Cn and φ = (φ 1,...,φ n ) a holomorphic self-map of Un. Let 0 ≤α 1. This paper shows that the composition operator C is bounded on the Lipschitz space Lip(Un) if and only if there exists M > 0 such that $$\sum\limits_{k,l = 1}^n {\left| {\frac{{\partial \phi _l }}{{\partial zk}}(z)} \right|\left( {\frac{{1 - \left| {z_k } \right|^2 }}{{1 - \left| {\phi _l (z)} \right|^2 }}} \right)^{1 - \alpha } } \leqslant M$$ for z ∈ Un. Moreover Cφ is compact on Lipα(Un) if and only if Cφ is bounded on Lipα(Un) and for every ε>0, there exists a δ > 0 such that $$\sum\limits_{k,l = 1}^n {\left| {\frac{{\partial \phi _l }}{{\partial zk}}(z)} \right|\left( {\frac{{1 - \left| {z_k } \right|^2 }}{{1 - \left| {\phi _l (z)} \right|^2 }}} \right)^{1 - \alpha } } \leqslant \varepsilon $$ whenever dist((z),?Un).展开更多
The authors prove a general Schwarz lemma at the boundary for holomorphic mappings from the polydisc to the unit ball in any dimensions.For the special case of one complex variable,the obtained results give the classi...The authors prove a general Schwarz lemma at the boundary for holomorphic mappings from the polydisc to the unit ball in any dimensions.For the special case of one complex variable,the obtained results give the classic boundary Schwarz lemma.展开更多
In this paper, we characterize the essential normality of quasi-homogeneous quotient Hardy modules over the polydisc by giving a complete criterion in terms of partially maximal ideals. Then we generalize this result ...In this paper, we characterize the essential normality of quasi-homogeneous quotient Hardy modules over the polydisc by giving a complete criterion in terms of partially maximal ideals. Then we generalize this result to the weighted Bergman modules.展开更多
基金This work was supported in part by the National Natural Science Foundation of China ( Grant No. 19871081).
文摘Let Un be the unit polydisc of ?n and φ=(φ1, ?, φ n ) a holomorphic self-map of Un. As the main result of the paper, it shows that the composition operator C is compact on the Bloch space β(Un) if and only if for every ε > 0, there exists a δ > 0, such that $$\sum\limits_{k,1 = 1}^n {\left| {\frac{{\partial \phi _l }}{{\partial z_k }}(z)} \right|} \frac{{1 - |z_k |^2 }}{{1 - |\phi _l (z)|^2 }}< \varepsilon ,$$ whenever dist(φ(z), ?U n )<δ.
基金This work was supported in part by the National Natural Science Foundation ofChina (Grant Nos. 19871081 & 10001030) LiuHui Center for Applied Mathematics, Nankai University and Tianjin University.
文摘Let Un be the unit polydisc of Cn and φ = (φ 1,...,φ n ) a holomorphic self-map of Un. Let 0 ≤α 1. This paper shows that the composition operator C is bounded on the Lipschitz space Lip(Un) if and only if there exists M > 0 such that $$\sum\limits_{k,l = 1}^n {\left| {\frac{{\partial \phi _l }}{{\partial zk}}(z)} \right|\left( {\frac{{1 - \left| {z_k } \right|^2 }}{{1 - \left| {\phi _l (z)} \right|^2 }}} \right)^{1 - \alpha } } \leqslant M$$ for z ∈ Un. Moreover Cφ is compact on Lipα(Un) if and only if Cφ is bounded on Lipα(Un) and for every ε>0, there exists a δ > 0 such that $$\sum\limits_{k,l = 1}^n {\left| {\frac{{\partial \phi _l }}{{\partial zk}}(z)} \right|\left( {\frac{{1 - \left| {z_k } \right|^2 }}{{1 - \left| {\phi _l (z)} \right|^2 }}} \right)^{1 - \alpha } } \leqslant \varepsilon $$ whenever dist((z),?Un).
基金supported by the National Science Foundation of China(Nos.11671361,11571256)
文摘The authors prove a general Schwarz lemma at the boundary for holomorphic mappings from the polydisc to the unit ball in any dimensions.For the special case of one complex variable,the obtained results give the classic boundary Schwarz lemma.
基金supported by National Natural Science Foundation of China (Grant Nos. 11471189 and 11501329)Shandong Province Natural Science Foundation (Grant No. ZR2014AQ009)the Fundamental Research Funds of Shandong University (Grant No. 2015GN017)
文摘In this paper, we characterize the essential normality of quasi-homogeneous quotient Hardy modules over the polydisc by giving a complete criterion in terms of partially maximal ideals. Then we generalize this result to the weighted Bergman modules.
