scholarly journals Holomorphic Functions of Slow Growth on Nested Covering Spaces of Compact Manifolds

2000 ◽  
Vol 52 (5) ◽  
pp. 982-998 ◽  
Author(s):  
Finnur Lárusson
Keyword(s):  
Riemann Surfaces ◽  
Bergman Spaces ◽  
Holomorphic Functions ◽  
Weighted Bergman Spaces ◽  
Projective Manifold ◽  
Important Special Case ◽  
Covering Group ◽  
Bounded Holomorphic Function ◽  
Domains Of Holomorphy ◽  
Bounded Holomorphic

AbstractLet Y be an infinite covering space of a projective manifold M in N of dimension n ≥ 2. Let C be the intersection with M of at most n − 1 generic hypersurfaces of degree d in N. The preimage X of C in Y is a connected submanifold. Let φ be the smoothed distance from a fixed point in Y in a metric pulled up from M. Let φ(X) be the Hilbert space of holomorphic functions f on X such that f2e−φ is integrable on X, and define φ(Y) similarly. Our main result is that (under more general hypotheses than described here) the restriction φ(Y) → φ(X) is an isomorphism for d large enough.This yields new examples of Riemann surfaces and domains of holomorphy in n with corona. We consider the important special case when Y is the unit ball in n, and show that for d large enough, every bounded holomorphic function on X extends to a unique function in the intersection of all the nontrivial weighted Bergman spaces on . Finally, assuming that the covering group is arithmetic, we establish three dichotomies concerning the extension of bounded holomorphic and harmonic functions from X to .

Local Boundary Behavior of Bounded Holomorphic Functions

1978 ◽  
Vol 30 (03) ◽  
pp. 583-592 ◽  
Author(s):  
Alexander Nagel ◽  
Walter Rudin
Keyword(s):  
Smooth Boundary ◽  
Boundary Behavior ◽  
Holomorphic Functions ◽  
Area Measure ◽  
Surface Area Measure ◽  
Smooth Curves ◽  
Local Boundary ◽  
Bounded Holomorphic Function ◽  
Bounded Holomorphic ◽  
Dimensional Lebesgue Measure

Let D ⊂⊂ Cn be a bounded domain with smooth boundary ∂D, and let F be a bounded holomorphic function on D. A generalization of the classical theorem of Fatou says that the set E of points on ∂D at which F fails to have nontangential limits satisfies the condition σ (E) = 0, where a denotes surface area measure. We show in the present paper that this result remains true when σ is replaced by 1-dimensional Lebesgue measure on certain smooth curves γ in ∂D. The condition that γ must satisfy is that its tangents avoid certain directions.

scholarly journals Composition followed by differentiation between weighted Bergman spaces and weighted Banach spaces of holomorphic functions

2014 ◽  
Vol 6 (1) ◽  
pp. 107-116
Author(s):  
Elke Wolf
Keyword(s):  
Banach Spaces ◽  
Composition Operator ◽  
Bergman Spaces ◽  
Holomorphic Functions ◽  
Differentiation Operator ◽  
Weighted Bergman Spaces ◽  
Open Unit ◽  
Open Unit Disk ◽  
Spaces Of Holomorphic Functions ◽  
Weighted Banach Spaces

AbstractLet Φ be an analytic self-map of the open unit disk D in the complex plane. Such a map induces through composition a linear composition operator CΦ: f ↦ f◦Φ.We are interested in the combination of CΦwith the differentiation operator D, that is in the operator DCΦ: f ↦ Φ` · (f ◦ Φ) acting between weighted Bergman spaces and weighted Banach spaces of holomorphic functions

Composition with a Nonhomogeneous Bounded Holomorphic Function on the Ball

1989 ◽  
Vol 41 (5) ◽  
pp. 870-881
Author(s):  
Jun Soo Choa ◽  
Hong Oh Kim
Keyword(s):  
Holomorphic Functions ◽  
Bounded Mean Oscillation ◽  
Open Unit ◽  
Homogeneous Polynomials ◽  
Open Unit Disc ◽  
Mean Oscillation ◽  
Bounded Holomorphic Function ◽  
Euclidean Unit Ball ◽  
Image Position ◽  
Bounded Holomorphic

For an integer n > 1, the letters U and Bn denote the open unit disc in C and the open euclidean unit ball in Cn, respectively. It is known that the homogeneous polynomialswhere bα is chosen so that , have the following pull-back property:If g ∈ ℬ(U) the Block space, then , the space of holomorphic functions on Bn of bounded mean oscillation, forand.

scholarly journals Addendum to the Paper “A note on Weighted bergman Spaces and the cesaro operator”

2005 ◽  
Vol 180 ◽  
pp. 77-90 ◽  
Author(s):  
Der-Chen Chang ◽  
Stevo Stević
Keyword(s):  
Bergman Space ◽  
Bergman Spaces ◽  
Holomorphic Functions ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Cesàro Operator ◽  
Integral Means ◽  
Measurable Functions ◽  
Unit Polydisk ◽  
Cesaro Operator

AbstractLet H(Dn) be the space of holomorphic functions on the unit polydisk Dn, and let , where p, q> 0, α = (α1,…,αn) with αj > -1, j =1,..., n, be the class of all measurable functions f defined on Dn such thatwhere Mp(f,r) denote the p-integral means of the function f. Denote the weighted Bergman space on . We provide a characterization for a function f being in . Using the characterization we prove the following result: Let p> 1, then the Cesàro operator is bounded on the space .

scholarly journals THE DIRECT INTEGRAL OF SOME WEIGHTED BERGMAN SPACES

2007 ◽  
Vol 50 (1) ◽  
pp. 115-122 ◽  
Author(s):  
Meng-Kiat Chuah
Keyword(s):  
Hilbert Space ◽  
Convex Function ◽  
Complex Space ◽  
Bergman Spaces ◽  
Geometric Quantization ◽  
Holomorphic Functions ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Weighted Bergman Spaces ◽  
Direct Integral

AbstractLet $G$ be the abelian Lie group $\mathbb{R}^n\times\mathbb{R}^k/\mathbb{Z}^k$, acting on the complex space $X=\mathbb{R}^{n+k}\times\ri G$. Let $F$ be a strictly convex function on $\mathbb{R}^{n+k}$. Let $H$ be the Bergman space of holomorphic functions on $X$ which are square-integrable with respect to the weight $e^{-F}$. The $G$-action on $X$ leads to a unitary $G$-representation on the Hilbert space $H$. We study the irreducible representations which occur in $H$ by means of their direct integral. This problem is motivated by geometric quantization, which associates unitary representations with invariant Kähler forms. As an application, we construct a model in the sense that every irreducible $G$-representation occurs exactly once in $H$.

Isometries of Weighted Bergman Spaces

1982 ◽  
Vol 34 (4) ◽  
pp. 910-915 ◽  
Author(s):  
Clinton J. Kolaski
Keyword(s):  
Bergman Space ◽  
Bergman Spaces ◽  
Holomorphic Functions ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Bounded Symmetric Domains ◽  
General Argument ◽  
Image Position ◽  
Linear Isometries

In [2], [8] and [10], Forelli, Rudin and Schneider described the isometries of the Hp spaces over balls and polydiscs. Koranyi and Vagi [6] noted that their methods could be used to describe the isometries of the Hp spaces over bounded symmetric domains. Recently Kolaski [4] observed that the algebraic techniques used above and Rudin's theorem on equimeasurability extended to the Bergman spaces over bounded Runge domains. In this paper we use the same general argument to characterize the onto linear isometries of the weighted Bergman spaces over balls and polydiscs, (all isometries referred to are assumed to be linear).2. Preliminaries. Horowitz [3] first defined the weighted Bergman space Ap,α(0 < p < ∞, 0 < α < ∞) to be the space of holomorphic functions f in the disc which satisfy(1)

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