Dirichlet-type Spaces Research Articles

Higher radial derivative of functions of [formula omitted] spaces and its applications

In this paper, we give some characterizations of the integral and Carleson type measure both associated with higher radial derivative for holomorphic functions of Q p , 0 and Q p spaces in the unit ball of C n . As applications, some sharp forms of the relations between Dirichlet type spaces and Q p , 0 spaces are shown by means of higher radial derivative, Carleson type measure and constructive methods for the cases of determinism and randomization, respectively.

Growth properties and sequences of zeros of analytic functions in spaces of Dirichlet type

AbstractFor 0 < p < ∞, we let pp−1 denote the space of those functions f that are analytic in the unit disc Δ = {z ∈ C: |z| < 1} and satisfy ∫Δ(1 – |z|)p−1|f′(z)|pdx dy < ∞ The spaces pp−1 are closely related to Hardy spaces. We have, p−1p ⊂ Hp, if 0 < p ≦ 2, and Hp ⊂ pp−1, if 2 ≦ p < ∞. In this paper we obtain a number of results about the Taylor coefficients of pp-1 -functions and sharp estimates on the growth of the integral means and the radial growth of these functions as well as information on their zero sets.2000 Mathematics subject classification: primary 30D35, 30D55, 46E15.

Some new characterizations of Dirichlet type spaces on the unit ball of [formula omitted

The Dirichlet type spaces on the unit ball of C n are characterized by some oscillations in the Bergman metric in this paper. As an application of these new characterizations of Dirichlet type spaces, we prove that composition operators induced by biholomorphic maps are bounded on D p ( − 1 ⩽ p < n ) .

Boundary behaviour of analytic functions in spaces of Dirichlet type

For and, we let be the space of all analytic functions in such that belongs to the weighted Bergman space. We obtain a number of sharp results concerning the existence of tangential limits for functions in the spaces. We also study the size of the exceptional set, where denotes the radial variation of along the radius, for functions.

Vector-valued dirichlet-type functions on the unit ball of C n

The vector-valued Dirichlet-type spaces on the unit ball of Cn is introduced. We discuss the pointwise multipliers of Dirichlet-type spaces. Sufficient conditions of the pointwise multipliers of\(D_\mu ^2 for 0 \leqslant \mu< 2 if n = 1 or D_{\mu .q}^2 for 0< \mu< 1 if n \geqslant 2\) are given. Finally, Rademacherp-type space is characterized by vector-valued sequence spaces.

Higher order Hilbert-Schmidt Hankel forms and tensors of analytical kernels

The symbols of $n^{\hbox{th}}$-order Hankel forms defined on the product of certain reproducing kernel Hilbert spaces $H(k_{i})$, $i=1,2$, in the Hilbert-Schmidt class are shown to coincide with the orthogonal complement in $H(k_{1})\otimes H(k_{2})$ of the ideal of polynomials which vanish up to order $n$ along the diagonal. For tensor products of weighted Bergman and Dirichlet type spaces (including the Hardy space) we introduce a higher order restriction map which allows us to identify the relative quotient of the $n^{\hbox{th}}$-order ideal modulo the $(n+1)^{\hbox{st}}$-order one as a direct sum of single variable Bergman and Dirichlet-type spaces. This generalizes the well understood $0^{\hbox{th}}$-order case.

An interpolation theorem for Hilbert spaces with Nevanlinna-Pick kernel

We prove an interpolation theorem for Hilbert spaces of analytic functions that have the Nevanlinna-Pick property. This result applies to Dirichlet and Dirichlet-type spaces, and in particular a short proof of the theorem by Marshall-Sundberg on interpolating sequences is obtained.

ON DIRICHLET TYPE SPACES AND α-BLOCH SPACES IN THE UNIT BALL OF Cn

In this paper, the authors study the inclusion relations between Dirichlet type spaces Dτ and α-Bloch spaces βα by means of higher radial derivative. The strictness and the best possibility of the inclusion relations are shown with constructive methods. Furthermore, they sharpen one of the results when τ = n, which proves that a conjecture in [7] is true.

Univalent functions, Hardy spaces and spaces of Dirichlet type

We prove that for $p\in (0,\infty)$ an analytic univalent function in the unit disk belongs to the Hardy space $H^p$ if and only if it belongs to the Dirichlet type space $\mathcal {D}_{p-1}^p$.

Small Hankel Operators on the Dirichlet-Type Spaces on the Unit Ball of C n

In this paper the small Hankel operators on the Dirichlet-type spaces Dp on the unit ball of C are considered. A similar result to that of the one-dimensional setting is given, which characterizes the boundedness of the small Hankel operators on Dp.

The pointwise multipliers of Bloch type space βp and Dirichlet type space Dq on the unit ball of Cn

In the paper, we will discuss the pointwise multipliers from Dirichlet type space D p to Bloch type space β q on the unit ball of C n . The multiplier spaces M( D p , β q ) are fully characterized for all p, q.

Multivariable Nehari problem and interpolation

In this paper we obtain a multivariable Nehari type theorem for Hankel operators, extending the classical result as well as Treil–Volberg generalization. The result is based on a new generalization of the noncommutative commutant lifting theorem which extends to several variables the commutant lifting result obtained by Treil and Volberg, and recently generalized by Biswas, Foiaş, and Frazho. An extension of I.S. Iokhvidov–Ky Fan theorem is obtained and used to prove a multivariable Adamian–Arov–Krein type theorem for generalized Hankel operators. As consequences, we obtain Carathéodory and Nevanlinna–Pick type interpolation results for “meromorphic” operators on Fock spaces (resp. operator-valued meromorphic functions on the unit ball of C n ). The Nehari type results are used to obtain descriptions of Hankel operators acting on Fock spaces or weighted Fock spaces (multivariable Dirichlet type spaces) and to solve new “weighted” interpolation problems (e.g. Sarason) for noncommutative analytic Toeplitz algebras, which have consequences to the operator-valued analytic interpolation in the unit ball of C n . Moreover, using central intertwining liftings, we obtain explicit formulas for certain solutions of the weighted interpolation problems. As another application of the generalized noncommutative commutant lifting theorem, we derive an abstract interpolation problem for multipliers. This is used to solve a left tangential Nevanlinna–Pick type interpolation problem with operatorial argument for noncommutative analytic Toeplitz algebras.

Multipliers and Carleson Measures for $ D(\mu) $

In this paper, the multipliers of the Dirichlet-type space \( D(\mu) \) associated with a positive measure \( \mu \) on the unit circle are characterized in terms of \( \mu\)-Carleson measures. A geometric description of \( \mu\)-Carleson measures is given for the cases of \( \mu \) being absolutely continuous with an \( A_{2} \) weight, and of \( \mu \) being a finite sum of atoms.

Gleason's problem and homogeneous interpolation in Hardy and Dirichlet-type spaces of the ball

We solve Gleason's problem in the reproducing kernel Hilbert spaces with reproducing kernels 1/(1−∑ 1 Nz j w j) r for real r>0 and their counterparts for r⩽0, and study the homogeneous interpolation problem in these spaces.

Complete Nevanlinna–Pick Property of Dirichlet-Type Spaces

We prove that all Dirichlet-type spaces of functions analytic in the unit disk whose derivatives are square area integrable with superharmonic weights have complete Nevanlinna–Pick reproducing kernels. As a corollary, we obtain a commutant lifting theorem for cyclic analytic two-isometries.

Global integral criteria for composition operators

Let Dα denote the Dirichlet-type space of functions analytic on the unit disk U and Qα the conformal invariant version of this space. Any analytic self-map φ of U induces a composition operator Cφ acting on Dα, respectively, Qα by Cφf=f∘φ, where f∈Dα, respectively, f∈Qα. The aim of this paper is to characterize boundedness and compactness of such operators in terms of global area integrals of φ.

A New Characterization of Dirichlet Type Spaces on the Unit Ball of Cn

In this paper a BMO-type characterization of Dirichlet type spaces Dp on the unit ball of Cn is given.

Multipliers on Dirichlet Type Spaces

In this paper, we characterize the pointwise multiplier space M (Dτ, Dμ) of Dirichlet type spaces in the unit ball of Cn for the values of τ, μ in three cases: (i) τ n, and construct two functions to show that M(Dτ) ⊂Dτ properly if τ≤n and M(Dτ) ⊂M(Dμ) properly if τ > μ and τ > n− 1.

Reproducing Kernels and Extremal Functions in Dirichlet-Type Spaces

Some identities are established for reproducing kernels and extremal functions in Dirichlet-type spaces. In certain cases, these identities are characteristic. A new proof of Aleman's theorem on extremal functions as contractive multipliers is obtained. In contrast to the original one, this proof can be extended to vector-valued spaces. Bibliography: 11 titles.

The ¯∂-problem for multipliers of the Sobolev space

For alpha is an element of (0, 1/2], let M(L-alpha(2)) and M(D-alpha) be the spaces of multipliers of L-alpha(2), the Sobolev space on the unit circle, and D-alpha, the Dirichlet type space on the ...