Harmonic Besov spaces on the ball


Gergün S., Kaptanoğlu H. T., Üreyen A. E.

INTERNATIONAL JOURNAL OF MATHEMATICS, vol.27, no.9, 2016 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 27 Issue: 9
  • Publication Date: 2016
  • Doi Number: 10.1142/s0129167x16500701
  • Journal Name: INTERNATIONAL JOURNAL OF MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Keywords: Spherical harmonic, zonal harmonic, Gegenbauer (ultraspherical) polynomial, Poisson kernel, reproducing kernel, radial fractional derivative, Mobius transformation, Bergman space, Besov space, Hardy space, Bergman projection, atomic decomposition, boundary growth, Fourier coefficient, duality, interpolation, Gleason problem, UNIT BALL, REPRODUCING KERNELS, TOEPLITZ-OPERATORS, BERGMAN SPACES, BLOCH, PROJECTIONS, DUALITY
  • Dokuz Eylül University Affiliated: Yes

Abstract

We initiate a detailed study of two-parameter Besov spaces on the unit ball of R-n 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.