THE REPRODUCING KERNEL OF H2 AND RADIAL EIGENFUNCTIONS OF THE HYPERBOLIC LAPLACIAN

In the paper we characterize the reproducing kernel K-n,K-h for the Hardy space H-2 of hyperbolic harmonic functions on the unit ball B in R-n. Specifically we prove that K-n,K-h (x, y) = Sigma(infinity)(alpha=0) S-n,S-alpha (vertical bar x vertical bar) S-n,S-alpha (vertical bar y vertical bar) Z(alpha) (x, y), where the series converges absolutely and uniformly on K x B for every compact subset K of B. In the above, S-n,S-alpha is a hypergeometric function and Z(alpha) is the reproducing kernel of the space of spherical harmonics of degree alpha. In the paper we prove that 0 <= K-n,K-h(x, y) <= C-n/(1 - 2 < x, y > + vertical bar x vertical bar(2)vertical bar y vertical bar(2))(n-1), where C-n is a constant depending only on n. It is known that the diagonal function K-n,K-h(x, x) is a radial eigenfunction of the hyperbolic Laplacian Delta(h) on B with eigenvalue lambda(2) = 8(n - 1)(2). The result for n = 4 provides motivation that leads to an explicit characterization of all radial eigenfunctions of Delta(h) on B. Specifically, if g is a radial eigenfunction of Delta(h) with eigenvalue lambda(alpha) = 4(n - 1)(2)alpha(alpha - 1), then g(r) = g(0) p(n,alpha)(r(2))/(1 - r(2))((alpha - 1)(n - 1)) , where p(n,alpha) is again a hypergeometric function. If alpha is an integer, then p(n,alpha)(r(2)) is a polynomial of degree 2(alpha - 1)(n - 1).