Reproducing kernels for harmonic Bergman spaces of the unit ball

1998 ◽  
Vol 125 (1) ◽  
pp. 25-35 ◽  
Author(s):  
Jie Miao
Keyword(s):  
Unit Ball ◽  
Bergman Spaces ◽  
Reproducing Kernels ◽  
Harmonic Bergman Spaces

scholarly journals Compact Toeplitz operators on weighted harmonic Bergman spaces

1998 ◽  
Vol 64 (1) ◽  
pp. 136-148 ◽  
Author(s):  
Karel Stroethoff
Keyword(s):  
Unit Ball ◽  
Harmonic Functions ◽  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Harmonic Bergman Spaces

AbstractWe consider the Bergman spaces consisting of harmonic functions on the unit ball in Rn that are squareintegrable with respect to radial weights. We will describe compactness for certain classes of Toeplitz operators on these harmonic Bergman spaces.

scholarly journals Harmonic Besov spaces on the ball

2016 ◽  
Vol 27 (09) ◽  
pp. 1650070 ◽  
Author(s):  
Seçil Gergün ◽  
H. Turgay Kaptanoğlu ◽  
A. Ersin Üreyen
Keyword(s):  
Besov Spaces ◽  
Bergman Spaces ◽  
Reproducing Kernels ◽  
Integral Representations ◽  
Kernel Estimates ◽  
Harmonic Bergman Spaces ◽  
Infinite Sums ◽  
Bergman Projections ◽  
Derivatives Of

We initiate a detailed study of two-parameter Besov spaces on the unit ball of [Formula: see text] consisting of harmonic functions whose sufficiently high-order radial derivatives lie in harmonic Bergman spaces. We compute the reproducing kernels of those Besov spaces that are Hilbert spaces. The kernels are weighted infinite sums of zonal harmonics and natural radial fractional derivatives of the Poisson kernel. Estimates of the growth of kernels lead to characterization of integral transformations on Lebesgue classes. The transformations allow us to conclude that the order of the radial derivative is not a characteristic of a Besov space as long as it is above a certain threshold. Using kernels, we define generalized Bergman projections and characterize those that are bounded from Lebesgue classes onto Besov spaces. The projections provide integral representations for the functions in these spaces and also lead to characterizations of the functions in the spaces using partial derivatives. Several other applications follow from the integral representations such as atomic decomposition, growth at the boundary and of Fourier coefficients, inclusions among them, duality and interpolation relations, and a solution to the Gleason problem.

scholarly journals Compact Hankel operators on weighted harmonic Bergman spaces

1997 ◽  
Vol 39 (1) ◽  
pp. 77-84 ◽  
Author(s):  
Karel Stroethoff
Keyword(s):  
Unit Ball ◽  
Harmonic Functions ◽  
Bergman Spaces ◽  
Hankel Operators ◽  
Weighted Bergman Spaces ◽  
Harmonic Bergman Spaces

AbstractWe prove the compactness of certain Hankel operators on weighted Bergman spaces of harmonic functions on the unit ball in Rn.

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

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.

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