Holomorphic Functions of Exponential Type and Duality for Stein Groups with Algebraic Connected Component of Identity

Hypercyclicity of composition operators in Stein manifolds

We characterise hypercyclic composition operators $C_φ:f\mapsto f\circφ$ on the space of functions holomorphic on $Ω$, where $Ω$ is a connected Stein manifold and $φ$ is a holomorphic self-mapping of $Ω$. In the case when all balls with respect to the Carathéodory pseudodistance are relatively compact in $Ω$, we show that much simpler characterisation is possible (many natural classes of domains in $\CC^N$ satisfy this condition). Moreover, we show that in such a class of manifolds, and in simply connected and infinitely connected planar domains, hypercyclicity of $C_φ$ implies its hereditary …

Manifolds of holomorphic mappings from strongly pseudoconvex domains

Let D be a bounded strongly pseudoconvex domain in a Stein manifold S and let Y be a complex manifold. We prove that the graph of any continuous map from the closure of D to Y which is holomorphic in D admits a basis of open Stein neighborhoods in S x Y. Using this we show that many classical spaces of maps from the closure of D to Y which are holomorphic in D are infinite dimensional complex manifolds which are modeled on locally convex topological vector spaces (Banach, Hilbert or Frechet).

极化 Hardy–Stein 恒等式

Polarized Hardy--Stein identity

本文提出 Bregman 余散度及相关积分形式，证明了向量函数与两个实函数典范配对情形下的极化 Hardy–Stein 恒等式。

Approximation of holomorphic mappings on strongly pseudoconvex domains

Let D be a relatively compact strongly pseudoconvex domain in a Stein manifold, and let Y be a complex manifold. We prove that the set A(D,Y), consisting of all continuous maps from the closure of D to Y which are holomorphic in D, is a complex Banach manifold. When D is the unit disc in C (or any other topologically trivial strongly pseudoconvex domain in a Stein manifold), A(D,Y) is locally modeled on the Banach space A(D,C^n)=A(D)^n with n=dim Y. Analogous results hold for maps which are holomorphic in D and of class C^r up to the boundary for any positive integer r. We also establish the O…

On holomorphic reflexivity conditions for complex Lie groups

We consider Akbarov's holomorphic version of the non-commutative Pontryagin duality for a complex Lie group. We prove, under the assumption that $G$ is a Stein group with finitely many components, that (1) the topological Hopf algebra of holomorphic functions on $G$ is holomorphically reflexive if and only if $G$ is linear; (2) the dual cocommutative topological Hopf algebra of exponential analytic functional on $G$ is holomorphically reflexive. We give a counterexample, which shows that the first criterion cannot be extended to the case of infinitely many components. Nevertheless, we conjectu…

Hardy-Stein identities and square functions for semigroups

We prove a Hardy-Stein type identity for the semigroups of symmetric, pure-jump Lévy processes. Combined with the Burkholder-Gundy inequalities, it gives the $L^p$ two-way boundedness, for $1<p<\infty$, of the corresponding Littlewood-Paley square function. The square function yields a direct proof of the $L^p$ boundedness of Fourier multipliers obtained by transforms of martingales of Lévy processes.

全纯函数空间的度量熵

Metric entropy of the space of holomorphic functions

研究复流形紧子集上全纯函数相关紧集的度量熵，通过确定其渐近性量化紧性，为\(\mathbb{C}^n\)中区域的柯尔莫哥洛夫问题提供新解并推广到任意斯坦流形区域。

Hausdorff Operators on Some Spaces of Holomorphic Functions on the Unit Disc

We introduce Hausdorff operators over the unit disc and give conditions for boundedness of such operator in Bloch, Bergman, and Hardy spaces on the disc. Identity approximation by Hausdorff operators is also considered.

Nonlinear nonlocal Douglas identity

We give Hardy-Stein and Douglas identities for nonlinear nonlocal Sobolev-Bregman integral forms with unimodal Lévy measures. We prove that the corresponding Poisson integral defines an extension operator for the Sobolev-Bregman spaces. We also show that the Poisson integrals are quasiminimizers of the Sobolev-Bregman forms.

半群的全纯泛函演算与向量值Littlewood-Paley-Stein理论

Holomorphic functional calculus and vector-valued Littlewood-Paley-Stein theory for semigroups

基于全纯泛函演算新方法，研究正则压缩半群的向量值Littlewood-Paley-Stein理论，导出了相关常数关于p的最优增长阶，并解决了Naor和Young关于经典Poisson和热半群最佳常数最优阶的猜想。