A CHARACTERISATION OF BERGMAN SPACES ON THE UNIT BALL OF ℂn (original) (raw)

Ann. Acad. Sci. Fenn. Math, 2008

We obtain several new characterizations for the standard weighted Bergman spaces A p α on the unit ball of C n in terms of the radial derivative, the holomorphic gradient, and the invariant gradient.

Bergman and Reinhardt weighted spaces of holomorphic functions

Illinois Journal of Mathematics

We study isometries between spaces of weighted holomorphic functions defined on bounded domains in C n. Using the Bergman kernel we see that it is possible to define a 'natural' weight on bounded domains in C n. We calculate the isometries of weighted spaces of holomorphic functions on the unit ball, the Thullen domains, the generalised Thullen domains and the domain with minimal complex norm.

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

Nagoya Mathematical Journal, 2005

LetH(Dn)be the space of holomorphic functions on the unit polydisk Dn, and let, wherep, q> 0,α = (α1,…,αn) with αj> -1,j =1,...,n, be the class of all measurable functions f defined on Dnsuch thatwhereMp(f,r)denote thep-integral means of the functionf. Denote the weighted Bergman space on. We provide a characterization for a functionfbeing in. Using the characterization we prove the following result: Letp> 1, then the Cesàro operator is bounded on the space.

The dual of a generalized weighted Bergman space

Advances in Operator Theory, 2020

The generalized weighted Bergman space HðB d ; kÞ is defined as a reproducing kernel Hilbert space of holomorphic functions on the open unit ball B d C d for all k [ 0. When k [ d, it is identical to the weighted Bergman space HL 2 ðB d ; l k Þ. We prove that the dual space HðB d ; aÞ Ã can be identified with another generalized weighted Bergman space HðB d ; bÞ under the pairing hf ; gi c ¼ R B d A k f ðzÞB k gðzÞ dl cþ2n ðzÞ; for f 2 HðB d ; aÞ; g 2 HðB d ; bÞ; where n ¼ d 2 AE Ç ; c ¼ aþb 2 and A k ; B k are operators related to the number operator N ¼ P d i¼1 z i o oz i :

Operators on weighted Bergman spaces (0

Duke Mathematical Journal, 1992

We describe the boundedness of a linear operator from B p (ρ) = {f : D → C analytic : D ρ(1 − |z|) (1 − |z|) |f (z)| p dA(z) 1/p < ∞} , for 0 < p ≤ 1 under some conditions on the weight function ρ, into a general Banach space X by means of the growth conditions at the boundary of certain fractional derivatives of a single X-valued analytic function. This, in particular, allows us to characterize the dual of B p (ρ) for 0 < p < 1 and to give a formulation of generalized Carleson measures in terms of the inclusion B 1 (ρ) ⊂ L 1 (D, µ). We then apply the result to the study of multipliers, Hankel operators and composition operators acting on B p (ρ) spaces.

Operators on weighted Bergman spaces

Contemporary Mathematics, 2006

Let ρ : (0, 1] → R + be a weight function and let X be a complex Banach space. We denote by A 1,ρ (D) the space of analytic functions in the disc D such that D |f (z)|ρ(1 − |z|)dA(z) < ∞ and by Bloch ρ (X) the space of analytic functions in the disc D with values in X such that sup |z|<1 1−|z| ρ(1−|z|) F (z) < ∞. We prove that, under certain assumptions on the weight, the space of bounded operators L(A 1,ρ (D), X) is isomorphic to Bloch ρ (X) and some applications of this result are presented. Several properties of generalized vector-valued Bloch functions are also considered.

Intersections and Unions of Weighted Bergman Spaces

2005

Certain intersection and unions with respect to different parameters of the weighted Bergman spaces are shown to be equal. These results are extended to a more general class of function spaces. Some of the results proved here play an important role in the study of complex linear differential equations in the unit disc.

Hyponormality on general Bergman spaces

Filomat, 2019

A bounded operator T on a Hilbert space is hyponormal if T*T-TT* is positive. We give a necessary condition for the hyponormality of Toeplitz operators on weighted Bergman spaces, for a certain class of radial weights, when the symbol is of the form f+g?, where both functions are analytic and bounded on the unit disk. We give a sufficient condition when f is a monomial.

Application of a Riesz-type formula to weighted Bergman spaces

Proceedings of the American Mathematical Society

Let D denote the unit disk in the complex plane. We consider a class of superbiharmonic weight functions w : D → R + whose growth are subject to the condition 0 ≤ w(z) ≤ C(1 − |z|) for some constant C. We first establish a Reisz-type representation formula for w, and then use this formula to prove that the polynomials are dense in the weighted Bergman space with weight w.

Bergman spaces with exponential type weights

Journal of Inequalities and Applications, 2021

For 1 ≤ p < ∞, let A p ω be the weighted Bergman space associated with an exponential type weight ω satisfying D K z (ξ) ω(ξ) 1/2 dA(ξ) ≤ Cω(z)-1/2 , z ∈ D, where K z is the reproducing kernel of A 2 ω. This condition allows us to obtain some interesting reproducing kernel estimates and more estimates on the solutions of the ∂-equation (Theorem 2.5) for more general weight ω *. As an application, we prove the boundedness of the Bergman projection on L p ω , identify the dual space of A p ω , and establish an atomic decomposition for it. Further, we give necessary and sufficient conditions for the boundedness and compactness of some operators acting from A p ω into A q ω , 1 ≤ p, q < ∞, such as Toeplitz and (big) Hankel operators.