Harmonic Besov spaces on the ball
Author
Gergün, S.
Kaptanoğlu, H. T.
Üreyen, A. E.
Date
2016Source Title
International Journal of Mathematics
Print ISSN
0129-167X
Electronic ISSN
1793-6519
Publisher
World Scientific Publishing
Volume
27
Issue
9
Pages
1650070-1 - 1650070-59
Language
English
Type
ArticleItem Usage Stats
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Abstract
We initiate a detailed study of two-parameter Besov spaces on the unit ball of ℝn 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. © 2016 World Scientific Publishing Company.
Keywords
atomic decompositionBergman projection
Bergman space
Besov space
Boundary growth
Duality
Fourier coefficient
Gegenbauer (ultraspherical) polynomial
Gleason problem
Hardy space
Interpolation
Möbius transformation
Poisson kernel
Radial fractional derivative
Reproducing kernel
Spherical harmonic
Zonal harmonic
31B05
31B10
31C25
26A33
33C55
42B35
45P05
46E22