Harmonic Besov spaces on the ball

Date
2016
Authors
Gergün, S.
Kaptanoğlu, H. T.
Üreyen, A. E.
Advisor
Instructor
Source 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
Article
Journal Title
Journal ISSN
Volume Title
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.

Course
Other identifiers
Book Title
Keywords
atomic decomposition, Bergman 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
Citation
Published Version (Please cite this version)