Quotients of the unit ball of ℂn for a free action of ℤ

LetB _n be the unit ball of ℂⁿ and ℤ ≅ Γ ⊂ AutB _n be generated by a parabolic element of AutB _n. We show that the quotientB _n/Γ is biholomorphic to a holomorphically convex domain of ℂⁿ, whose automorphism group is explicity described. It follows thatB _n/ℤ is Stein for any free action of ℤ. Investigation partially supported by University of Bologna. Funds for selected research topics. The second author was supported by an Instituto Nazionale di Alta Matematica grant.     

Schwarz-Pick estimates for bounded holomorphic functions in the unit ball of ℂ n

We give a Schwarz-Pick estimate for bounded holomorphic functions on unit ball in Cn, and generalize some early work of Schwarz-Pick estimates for bounded holomorphic functions on unit disk in C.     

Bounded projections and duality on spaces of holomorphic functions in the unit ball ofℂ n

In this paper three Banach spacesA ₀(),A andA ¹() of functions holomorphic in the unit ballB of ⁿ are defined. We exhibit bounded projections fromC ₀(B) ontoA ₀(), fromL ¹(B) ontoA ¹(), and fromL(B) ontoA(). Using these projections, we show thatA ₀()* A ¹() andA ¹()* A().Supported in part by the National Natural Science Foundation of China.     

Continuous embeddings in harmonic mixed norm spaces on the unit ball in ? n

In this paper continuous embeddings in spaces of harmonic functions with mixed norm on the unit ball in ? ⁿ are established, generalizing some Hardy-Littlewood embeddings for similar spaces of holomorphic functions in the unit disc. Differences in indices between the spaces of harmonic and holomorphic spaces are revealed. As a consequence an analogue of classical Fejér-Riesz inequality is obtained. Embeddings in the special case of Riesz systems are also established.     

Zonal spherical function on the Lie ball in ℂn

In this paper, we give the explicit formula for the zonal spherical function on the Lie ball in ?ⁿ.     

Randomization of Q_p spaces on the unit ball of C~n

In this paper, we study the random power series of several complex variables, and give some sufficient conditions for random power series to belong to Qp and Qp,0 spaces by means of their characterizations of Carleson measure. Moreover, we show that such conditions are best possible in some sense.     

A Trudinger–Moser inequality for a conical metric in the unit ball

Archiv der Mathematik - In this note, we prove a Trudinger–Moser inequality for a conical metric in the unit ball. Precisely, let $${\mathbb {B}}$$ be the unit ball in $${\mathbb {R}}^N$$...     

Bloch constant of holomorphic mappings on the unit polydisk of C~n

In this paper,we give a definition of Bloch mappings defined in the unit polydisk D~n, which generalizes the concept of Bloch functions defined in the unit disk D.It is known that Bloch theorem fails unless we have some restrictive assumption on holomorphic mappings in several complex variables.We shall establish the corresponding distortion theorems for subfamiliesβ(K)andβ_(loc)(K) of Bloch mappings defined in the polydisk D~n,which extend the distortion theorems of Liu and Minda to higher dimensions.As an application,we obtain lower and upper bounds of Bloch constants for various subfamilies of Bloeh mappings defined in D~n.In particular,our results reduce to the classical results of Ahlfors and Landau when n=1.     

A lower bound on the Kobayashi metric near a point of finite type in ℂ n

Let Ω be a bounded domain in ? ⁿ andbΩ smooth pseudoconvex near z₀ ∈bΩ of finite type. Then there are constantsc>0 and ε′>0 such that the Kobayashi metric,K _Ω(z; X), satisfiesK _Ω(z; X)≥c|X|δ(z)^?t′ for allX ∈T _z ^1,0 ? ⁿ in a neighborhood ofz ₀. Here δ(z) denotes the distance fromz tobΩ. As an application, we prove the Hölder continuity of proper holomorphic maps onto pseudoconvex domains.     

A class of normal weighted composition operators on the Fock space of ℂ n

This paper characterizes a class of normal weighted composition operators and their spectrum on the Fock space of Cⁿ.     

On the injectivity of twisted spherical means on ℂ n

In this paper we study the injectivity and injectivity sets for the twisted spherical means on ℂ ⁿ . These results are then used to study the same problems on the reduced Hiesenberg group.     

Characterizations of α-Bloch functions on the unit ball without use of derivative

In this paper we establish characterizations of α-Bloch functions on the unit ball without use of derivative, which are stronger, more precise and general than those obtained by Nowak and Zhao.     

Distortion theorems for subclasses of starlike mappings along a unit direction in ℂ

In this paper, we obtain a distortion theorem of Jacobian matrix for biholomorphic subclasses of starlike mappings along a unit direction on the unit polydisc. These results extend the classical distortion theorem of starlike mappings to higher dimensions. We then give an upper bound estimate for biholomorphic subclasses of starlike mappings along a unit direction in a complex Banach space.     

Analog of a theorem of forelli for boundary values of holomorphic functions on the unit ball of ?n

Let f ∈ C ^ω (∂B ⁿ ), where B ⁿ is the unit ball of ℂ ⁿ . We prove that if a,b ? [`(B)] ⁿa,b in {overline B ^n}, a ≠ b, for every complex line L passing through one of a or b, the restricted function f|_{L ??Bⁿ}f{|_{L cap partial {B^n}}} has a holomorphic extention to the cross-section L∩B ⁿ , then f is the boundary value of a holomorphic function in B ⁿ .