New atomic decompositions for Bergman spaces on the unit ball

2017 ◽  
Vol 66 (1) ◽  
pp. 205-235 ◽  
Author(s):  
Jens G Christensen ◽  
Karlheinz Grochenig ◽  
Gestur Olafsson
Keyword(s):  
Unit Ball ◽  
Bergman Spaces ◽  
Atomic Decompositions

О НЕКОТОРЫХ НОВЫХ ТЕОРЕМАХ ДЕКОМПОЗИЦИИ АНАЛИТИЧЕСКИХ МУЛЬТИФУНКЦИНАЛЬНЫХ ПРОСТРАНСТВ ГЕРЦА И БЕРГМАНА В ЕДИНИЧНОМ ШАРЕ И В ОГРАНИЧЕННЫХ ПСЕВДОВЫПУКЛЫХ ОБЛАСТЯХ В Cn

2020 ◽  
pp. 42-58
Author(s):  
E.B. Tomashevskaya ◽  
R.F. Shamoyan
Keyword(s):  
Unit Ball ◽  
Bergman Spaces ◽  
Integral Condition ◽  
Pseudoconvex Domains ◽  
Atomic Decompositions ◽  
Decomposition Theorems

Under certain integral condition which vanishes in onefunctional case we provide new sharp decomposition theorems for multifunctional Herz and Bergman spaces in the unit ball and pseudoconvex domains expanding known results from the unit ball. Our theorems extend also in various directions some known theorems on atomic decompositions of onefunctional Bergman spaces in the unit ball and in bounded pseudoconvex domains. Приведены новые теоремы декомпозиции для аналитических многофункциональных пространств Герца и Бергмана в единичном шаре и в ограниченных строго псевдовыпуклых областях в Cn, обобщающие некоторые ранее известные результаты для многофункциональных аналитических пространств Бергмана. Эти теоремы также обобщают в различных направлениях некоторые известные ранее результаты об атомическом разложении классических аналитических однофункциональных пространств Бергмана в единичном шаре и в ограниченных псевдовыпуклых областях в Cn.

A Characterization of Bergman Spaces on the Unit Ball of ℂn. II

2012 ◽  
Vol 55 (1) ◽  
pp. 146-152 ◽  
Author(s):  
Songxiao Li ◽  
Hasi Wulan ◽  
Kehe Zhu
Keyword(s):  
Holomorphic Function ◽  
Unit Ball ◽  
Bergman Space ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Double Integrals

AbstractIt has been shown that a holomorphic function f in the unit ball of ℂn belongs to the weighted Bergman space , p > n + 1 + α, if and only if the function | f(z) – f(w)|/|1 – 〈z, w〉| is in Lp( × , dvβ × dvβ), where β = (p + α – n – 1)/2 and dvβ(z) = (1 – |z|2)βdv(z). In this paper we consider the range 0 < p < n + 1 + α and show that in this case, f ∈ (i) if and only if the function | f(z) – f(w)|/|1 – hz, wi| is in Lp( × , dvα × dvα), (ii) if and only if the function | f(z)– f(w)|/|z–w| is in Lp( × , dvα × dvα). We think the revealed difference in the weights for the double integrals between the cases 0 < p < n + 1 + α and p > n + 1 + α is particularly interesting.

scholarly journals Further properties of the Bergman spaces of slice regular functions

2015 ◽  
Vol 15 (4) ◽  
Author(s):  
Fabrizio Colombo ◽  
J. Oscar González-Cervantes ◽  
Irene Sabadini
Keyword(s):  
Unit Ball ◽  
Half Space ◽  
Half Plane ◽  
Bergman Kernel ◽  
Bergman Spaces ◽  
Reflection Principle ◽  
Slice Regular Functions ◽  
Regular Functions ◽  
Complex Half Plane ◽  
Schwarz Reflection Principle

AbstractWe continue the study of Bergman theory for the class of slice regular functions. In the slice regular setting there are two possibilities to introduce the Bergman spaces, that are called of the first and of the second kind. In this paperwe mainly consider the Bergman theory of the second kind, by providing an explicit description of the Bergman kernel in the case of the unit ball and of the half space. In the case of the unit ball, we study the Bergman-Sce transform. We also show that the two Bergman theories can be compared only if suitableweights are taken into account. Finally,we use the Schwarz reflection principle to relate the Bergman kernel with its values on a complex half plane.

scholarly journals On boundedness of the weighted Bergman projections on the Lipschitz spaces

2002 ◽  
Vol 66 (3) ◽  
pp. 385-391
Author(s):  
Hong Rae Cho ◽  
Jinkee Lee
Keyword(s):  
Unit Ball ◽  
Bergman Spaces ◽  
Lipschitz Spaces ◽  
Bergman Projections

In this paper we study the boundedness of the weighted Bergman projections on the weighted subspaces of Bergman spaces and the Lipschitz spaces on the unit ball and the unit polydisc.

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