The reproducing kernel of H2 and radial eigenfunctions of the hyperbolic Laplacian

• Manfred Stoll ^[1]
  1. [1] University of south Carolina (Estados Unidos)
• Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 124, Nº 1, 2019, págs. 81-101
• Idioma: inglés
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• Resumen

  • In the paper we characterize the reproducing kernel Kn,h for the Hardy space H2 of hyperbolic harmonic functions on the unit ball B in Rn. Specifically we prove that Kn,h(x,y)=∑α=0∞Sn,α(|x|)Sn,α(|y|)Zα(x,y), where the series converges absolutely and uniformly on K×B for every compact subset K of B. In the above, Sn,α is a hypergeometric function and Zα is the reproducing kernel of the space of spherical harmonics of degree α. In the paper we prove that 0≤Kn,h(x,y)≤Cn(1−2⟨x,y⟩+|x|2|y|2)n−1, where Cn is a constant depending only on n. It is known that the diagonal function Kn,h(x,x) is a radial eigenfunction of the hyperbolic Laplacian Δh on B with eigenvalue λ2=8(n−1)2. The result for n=4 provides motivation that leads to an explicit characterization of all radial eigenfunctions of Δh on B. Specifically, if g is a radial eigenfunction of Δh with eigenvalue λα=4(n−1)2α(α−1), then g(r)=g(0)pn,α(r2)(1−r2)(α−1)(n−1), where pn,α is again a hypergeometric function. If α is an integer, then pn,α(r2) is a polynomial of degree 2(α−1)(n−1