The reproducing kernel thesis for lower bounds of weighted composition operators

It is shown that the property of being bounded below (having closed range) of weighted composition operators on Hardy and Bergman spaces can be tested by their action on a set of simple test functions, including reproducing kernels. The methods used in the analysis are based on the theory of reverse Carleson embeddings.     

An Integral Formula for Weighted Bergman Reproducing Kernels

An integral formula is obtained for reproducing kernels in weighted Bergman spaces with radial and logarithmically subharmonic weights in the unit disk. We deduce from it that these reproducing kernels have a special structure leading to the contractive divisor property of extermal functions.     

Bicomplex Weighted Hardy Spaces and Bicomplex <Emphasis Type="Italic">C</Emphasis>*-algebras

In this paper we study the bicomplex version of weighted Hardy spaces. Further, we describe reproducing kernels for the bicomplex weighted Hardy spaces. In particular, we generalize some results which holds for the classical weighted Hardy spaces. We also introduce the notion of bicomplex C*-algebra and discuss some of its properties.     

Reproducing Kernels and Conformal Mappings in

In this paper we look at the theory of reproducing kernels for spaces of functions in a Clifford algebra _{0, n}. A first result is that reproducing kernels of this kind are solutions to a minimum problem, which is a non-trivial extension of the analogous property for real and complex valued functions. In the next sections we restrict our attention to Szegö and Bergman modules of monogenic functions. The transformation property of the Szegö kernel under conformal transformations is proved, and the Szegö and Bergman kernels for the half space are calculated.     

Littlewood–Paley Theory and Function Spaces with Alocp Weights

Characterizations via convolutions with smooth compactly supported kernels and other distinguished properties of the weighted Besov–Lipschitz and Triebel–Lizorkin spaces on ℝⁿ with weights that are locally in A_p but may grow or decrease exponentially at infinity are investigated. Square–function characterizations of the weighted L^p and Hardy spaces with the above class of weights are also obtained. A certain local variant of the Calderón reproducing formula is constructed and widely used in the proofs.     

Linear methods for approximation of some classes of holomorphic functions from the Bergman space

We construct a linear method {ie910-01} for the approximation (in the unit disk) of classes of holomorphic functions {ie910-02} that are the Hadamard convolutions of the unit balls of the Bergman space A _p with reproducing kernels {ie910-03}. We give conditions for ψ under which the method {ie910-04} approximates the class {ie910-05} in the metrics of the Hardy space H _s and the Bergman space A _s , 1 ≤ s ≤ p, with an error that coincides in order with the value of the best approximation by algebraic polynomials. Translated from in Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 783–795, June, 2008.     

Reproducing kernels for harmonic Bergman spaces of the unit ball

We study reproducing kernels for harmonic Bergman spaces of the unit ball inR ⁿ . We establish some new properties for the reproducing kernels and give some applications of these properties.     

Generalized Interpolation in Bergman Spaces and Extremal Functions

In a recent paper A. Schuster and K. Seip [SchS] have characterized interpolating sequences for Bergman spaces in terms of extremal functions (or canonical divisors). As these are natural analogues in Bergman spaces of Blaschke products, this yields a Carleson type condition for interpolation. We intend to generalize this idea to generalized free interpolation in weighted Bergman spaces B^{p, α} as was done by V. Vasyunin [Va1] and N. Nikolski [Ni1] (cf.also [Ha2]) in the case of Hardy spaces. In particular we get a strong necessary condition for free interpolation in B^{p, α} on zero–sets of B^{p, α}–functions that in the special case of finite unions of B^{p, α}–interpolating sequences turns out to be also sufficient.     

Single-point extremal functions in weighted bergman spaces

For weighted Berman spaces in the unit disk the extremal functions for invariant subspaces formed by functions vanishing at a fixed point are studied. In the case where the weight is radial and logarithmically subharmonic, it is shown that such extremal functions can serve for the separation of single zeros. It is also proved that the reproducing kernels of the Bergman spaces are univalent functions. Bibliography: 7 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15, 1995, pp. 241–252.     

Reproducing kernels and contractive divisors in bergman spaces

In the Hardy spaces H^p of holomorphic functions, Blaschke products are applied to factor out zeros. However, for Bergman spaces, the zero sets of which do not necessarily satisfy the Blaschke condition, the study of divisors is a more recent development. Hedenmalm proved the existence of a canonical contractive zero-divisor which plays the role of a Blascke product in the Bergman space . Duren, Khavinson, Shapiro, and Sundberg later extended Hedenmalm's result to , 0<p<∞. In this paper, an explicit formula for the contractive divisor is given for a zero set that consists of two points of arbitrary multiplicities. There is a simple one-to-one correspondence between contractive divisors and reproducing kernels for certain weighted Bergman spaces. The divisor is obtained by calculating the associated reproducing kernel. The formula is then applied to obtain the contractive divisor for a certain regular zero set, as well as the contractive divisor associated with an inner function that has singular support on the boundary. Bibliography: 13 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 174–198.     

Best Linear Approximation Methods for Some Classes of Analytic Functions on the Unit Disk

Considering Banach Hardy spaces and weighted Bergman spaces, we find the sharp values of the Bernstein, Kolmogorov, Gelfand, and linear n-widths for the classes of analytic functions on the unit disk whose moduli of continuity of the rth derivatives averaged with weight are majorized by a given function satisfying some constraints.

     

Adjoints of rationally induced weighted composition operators on the Hardy,Bergman and Dirichlet spaces

We determine the adjoint of a multiplication operator with rational symbol u acting on various spaces of analytic functions, in which the denominator of u is a product of distinct linear factors. We use the results to represent the adjoints of weighted composition operators with rational symbols on the Hardy, Bergman and Dirichlet spaces.     

Hermitian Weighted Composition Operators and Bergman Extremal Functions

Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, Hermitian weighted composition operators on weighted Hardy spaces of the unit disk are studied. In particular, necessary conditions are provided for a weighted composition operator to be Hermitian on such spaces. On weighted Hardy spaces for which the kernel functions are ${(1 - \overline{w}z)^{-\kappa}}$ for κ ≥ 1, including the standard weight Bergman spaces, the Hermitian weighted composition operators are explicitly identified and their spectra and spectral decompositions are described. Some of these Hermitian operators are part of a family of closely related normal weighted composition operators. In addition, as a consequence of the properties of weighted composition operators, we compute the extremal functions for the subspaces associated with the usual atomic inner functions for these weighted Bergman spaces and we also get explicit formulas for the projections of the kernel functions on these subspaces.     

Isometries of some Banach spaces of analytic functions

We characterize the surjective isometries of a class of analytic functions on the disk which include the Analytic Besov spaceB _p and the Dirichlet spaceD ^p . In the case ofB _p we are able to determine the form of all linear isometries on this space. The isometries for these spaces are finite rank perturbations of integral operators. This is in contrast with the classical results for the Hardy and Bergman spaces where the isometries are represented as weighted compositions induced by inner functions or automorphisms of the disk.     

Resolvent estimates and decomposable extensions of generalized Cesàro operators

We determine the spectrum of generalized Cesàro operators with essentially rational symbols acting on various spaces of analytic functions, including Hardy spaces, weighted Bergman and Dirichlet spaces. Then we show that in all cases these operators are subdecomposable.     

Compactness of the Hardy-Littlewood operator on some spaces of harmonic functions

We study the compactness of the Hardy-Littlewood operator on several spaces of harmonic functions on the unit ball in ? ⁿ such as: a-Bloch, weighted Hardy, weighted Bergman, Besov, BMO _p , and Dirichlet spaces.     

Vector-Valued Kernels of Bergman Type

Bergman reproducing integral formulas can be obtained for holomorphic mappings \(f{:}\,{\mathbb {B}}\rightarrow {\mathbb {C}}^n,\,{\mathbb {B}}\) the open unit ball of \({\mathbb {C}}^n\), by applying the well-known formulas for scalar-valued functions on \({\mathbb {B}}\) to each coordinate function of f, provided those coordinate functions each lie in an appropriate Bergman space. Here, we consider an alternative formulation whereby f is reproduced as the integral of the product of a fixed vector-valued kernel and the scalar expression \(\langle f(z),z \rangle ,\,z\in {\mathbb {B}}\), where \(\langle \cdot ,\cdot \rangle \) is the Hermitian inner product in \({\mathbb {C}}^n\). We provide two different classes of vector-valued kernels that reproduce holomorphic mappings lying in spaces properly containing the weighted vector-valued Bergman spaces. An analysis of these larger spaces is given. The first set of kernels arises naturally from the scalar-valued Bergman kernels, while the second yields the orthogonal projection onto an isomorphic space of scalar-valued functions in the unweighted case.     

On the Libera operator

We show that the Libera operator, L, on some spaces of analytic functions is a continuous extension of the conjugate of the Cesàro operator. Results on L acting on various spaces are obtained. In particular, L maps the Bloch space into BMOA. We also prove some results on the best approximation by polynomials in Hardy and Bergman spaces.     

Toeplitz operators and weighted Bergman kernels

For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the reproducing kernels of Sobolev spaces of holomorphic functions of any real order. This generalizes the classical result of Fefferman for the unweighted Bergman kernel. Finally, we also exhibit a holomorphic continuation of the kernels with respect to the Sobolev parameter to the entire complex plane. Our main tool are the generalized Toeplitz operators of Boutet de Monvel and Guillemin.