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      Harmonic Besov spaces on the ball

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      Author
      Gergün, S.
      Kaptanoğlu, H. T.
      Üreyen, A. E.
      Date
      2016
      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
      Item Usage Stats
      133
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      111
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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 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
      Permalink
      http://hdl.handle.net/11693/37230
      Published Version (Please cite this version)
      http://dx.doi.org/10.1142/S0129167X16500701
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      • Department of Mathematics 644
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