Bers Space Research Articles

The Bers spaces and sequence spaces

The Bers spaces and sequence spaces

Weighted Bloch, Lipschitz, Zygmund, Bers, and Growth Spaces of the Ball: Bergman Projections and Characterizations

We determine precise conditions for the boundedness of Bergman projections from Lebesgue classes onto the spaces in the title, which are members of the same one-parameter family of spaces. The projections provide integral representations for the functions in the spaces. We obtain many properties of the spaces as straightforward corollaries of the projections, integral representations, and isometries among the spaces. We solve the Gleason problem and an extremal problem for point evaluations in each space. We establish maximality of these spaces among those that exhibit Mobius-type invariances ¨ and possess decent functionals. We find new Hermitian non-Kahlerian metrics ¨ that characterize half of these spaces by Lipschitz-type inequalities.

Products of composition and differentiation operators from Zygmund spaces to Bloch spaces and Bers spaces

The boundedness and compactness of the products of differentiation and composition operators from Zygmund spaces to Bloch spaces and Bers spaces are discussed in this paper.

Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces

Motivated by the recent paper [X. Zhu, Products of differentiation composition and multiplication from Bergman type spaces to Bers spaces, Integral Transform. Spec. Funct. 18 (3) (2007) 223–231], we study the boundedness and compactness of the weighted differentiation composition operator D φ , u n ( f ) ( z ) = u ( z ) f ( n ) ( φ ( z ) ) , where u is a holomorphic function on the unit disk D , φ is a holomorphic self-map of D and n ∈ N 0 , from the mixed-norm space H( p, q, ϕ), where p, q > 0 and ϕ is normal, to the weighted-type space H μ ∞ or the little weighted-type space H μ , 0 ∞ . For the case of the weighted Bergman space A α p , p > 1, some bounds for the essential norm of the operator are also given.

Harmonic maps and asymptotic Teichmüller space

In this paper, the asymptotic boundary behavior of a Hopf differential or the Beltrami coefficient of a harmonic map is investigated and certain compact properties of harmonic maps are established. It is shown that, if f is a quasiconformal harmonic diffeomorphism between two Riemann surfaces and is homotopic to an asymptotically conformal map modulo boundary, then f is asymptotically conformal itself. In addition, we prove that the harmonic embedding map from the Bers space B Q D (X) of an arbitrary hyperbolic Riemann surface X to the Teichmuller space T (X) induces an embedding map from the asymptotic Bers space A B Q D (X), a quotient space of B Q D (X), into the asymptotic Teichmuller space AT (X).

On the zeros of functions in the Bers space

We present some results on the distribution of zeros of functions in the Bers space Q(D), showing how the distribution depends on the bounds of the growth of │ƒ(z)│ HS │z│ƒ → 1, for ƒ Є Q(D). We also exhibit an open and dense subset, M C Q(D), which has the property of uniform control over the number of zeros in disks of hyperbolic radius l containes in D.

On Some Projection Operators Between Function Spaces

The paper deals with some projection operators from the space of essentially bounded functions to the Bers spaces of analytic functions. The results turn out to be closely related to some important topics in quasiconformal Teichmüller theory.

Uniformly bounded maximal $\varphi$-disks, Bers space and harmonic maps

We characterize Bers space by means of maximal ϕ-disks. As an application we show that the Hopf differential of a quasiregular harmonic map with respect to strongly negatively curved metric belongs to Bers space. Also we give further sufficient or necessary conditions for a holomorphic function to belong to Bers space.

Poincaré series and teichmüller space

We consider the Poincare series operator Θ as a holomorphic mapping of the Teichmuller space T(Γ) of a Fuchsian group Γ into the Bers space A(Γ) of integrable holomorphic quadratic forms, and show that if Γ is finitely generated and of the first kind, then ⊖ gives a local coordinate of T(Γ) at the origin. Conversely, if Θ(T(Γ)) contains an interior point in A(Γ), then Γ is finitely generated and of the first kind.

Order of approximation by electrostatic fields due to electrons

LetD be a Jordan domain in the complex plane andA q (D) the Bers space with norm ∥ ∥ q . IfD is the unit disk, it is known that ∥S n 0∥2≥π/18, whereS n =∑ k=1 n l/(z−z nk ) withz nk ∈∂D, so that approximation in ∥ ∥ q ,q<-2, is not possible. In this paper, we give an order of estimate of ∥f−S n ∥ q for 2<q<∞ when ∂D is a sufficiently smooth Jordan curve, and prove that this order of approximation is in general best possible.

The kernel of the Poincaré series operator

By modifying a proof of Ljan, a natural basis for the kernel of the Poincaré series operator in the Bers space can be given. The basic idea behind the proof also extends to give such a basis in the case of a general Kleinian group and a discontinuous group acting on certain domains in C n {{\mathbf {C}}^n} .

Polynomial density in Bers spaces

Let D be a bounded Jordan domain such that ∫ ∫ D λ D 2 − q d x d y &gt; ∞ \smallint \;\smallint {\;_D}\lambda _D^{2 - q}\;dx\;dy\; &gt; \infty for q &gt; 1 q &gt; 1 . Here λ D ( z ) {\lambda _D}(z) is the Poincaré metric for D. Define A q p ( D ) A_q^p(D) , the Bers space, to be the Fréchet space of holomorphic functions f on D, such that ‖ f ‖ q , p p = ∫ ∫ D λ D 2 − q p | f | p d x d y \left \| f \right \|_{q,p}^p = \smallint \;\smallint {\;_D}\lambda _D^{2 - qp}|f{|^p}\;dx\;dy is finite, 0 &gt; p &gt; ∞ , q p &gt; 1 0 &gt; p &gt; \infty ,qp &gt; 1 . It is well known that the polynomials are dense in A q p ( D ) A_q^p(D) for q p ⩾ 2 qp \geqslant 2 . We show that they are dense in A q p ( D ) A_q^p(D) for q p &gt; 1 qp &gt; 1 irrespective whether the boundary of D is rectifiable or not.

Polynomial Density in Bers Spaces

Let

On polynomial density in 𝐴_{𝑞}(𝐷)

Let D D be a bounded Jordan domain. Define A q ( D ) {A_q}(D) , the Bers space, to be the Banach space of holomorphic functions on D D , such that ∬ D | f | λ D 2 − q d x d y \iint _D {|f|\lambda _D^{2 - q}dxdy} is finite, where λ D ( z ) {\lambda _D}(z) is the Poincaré metric for D D . It is well known that the polynomials are dense in A q ( D ) {A_q}(D) for 2 ≦ q &gt; ∞ 2 \leqq q &gt; \infty and we shall prove they are dense in A q ( D ) {A_q}(D) for 1 &gt; q &gt; 2 1 &gt; q &gt; 2 if the boundary of D D is rectifiable. Also some remarks are made in case the boundary of D D is not rectifiable.

On Approximation in the Bers Spaces

Let $D$ be a Jordan domain in the complex plane with rectifiable boundary $C$. Let ${A_q}(D)$ denote the Bers space with norm $||\;|{|_q}$. We prove that if $f \in {A_q}(D),2 < q < \infty$, then there exist functions ${s_n}(z) = \Sigma _{k = 1}^n1/(z - {z_{n,k}}),\;{z_{n,k}} \in C{\text { for }}k = 1, \cdots ,n$, such that $||{s_n} - f|{|_q} \to 0$. This result does not hold for $1 < q \leqq 2$ even when $D$ is a disc.

On approximation in the Bers spaces

Let D D be a Jordan domain in the complex plane with rectifiable boundary C C . Let A q ( D ) {A_q}(D) denote the Bers space with norm | | | | q ||\;|{|_q} . We prove that if f ∈ A q ( D ) , 2 &gt; q &gt; ∞ f \in {A_q}(D),2 &gt; q &gt; \infty , then there exist functions s n ( z ) = Σ k = 1 n 1 / ( z − z n , k ) , z n , k ∈ C for k = 1 , ⋯ , n {s_n}(z) = \Sigma _{k = 1}^n1/(z - {z_{n,k}}),\;{z_{n,k}} \in C{\text { for }}k = 1, \cdots ,n , such that | | s n − f | | q → 0 ||{s_n} - f|{|_q} \to 0 . This result does not hold for 1 &gt; q ≦ 2 1 &gt; q \leqq 2 even when D D is a disc.