Harmonic Hadamard Manifolds and Gauss Hypergeometric Differential Equations

Abstract

A new class of harmonic Hadamard manifolds, those spaces called of hypergeometric type, is defined in terms of Gauss hypergeometric equations. Spherical Fourier transform defined on a harmonic Hadamard manifold of hypergeometric type admits an inversion formula. A characterization of harmonic Hadamard manifold being of hypergeometric type is obtained with respect to volume density.Mathematics Subject Classification (2010). Primary 53C21; Secondary 43A90, 42B10.

Harmonic manifolds of hypergeometric type and spherical Fourier transform

“…We show the Plancherel theorem which asserts that H is isometric with respect to certain inner products by applying the inversion formula which was obtained by the authors in [21] together with the convolution rule, valid for simply connected harmonic Hadamard manifolds of Q > 0. The convolution rule is shown in Theorem 9.2.…”

“…The proof of the inversion formula of non-spherical Fourier transform is there based on the spherical inversion formula, whose proof is given by applying the notion of hypergroup structure, a notion in Fourier analysis. In our setting, however, as indicated above and also in [21] the proof is indebted to a simple geometric argument and a basic theorem in analysis.…”

“…Further the inequalities We proved in [21] the inversion formula for the spherical Fourier transform by utilizing the Green's formula for the Laplace-Beltrami operator ∆ together with the Riemann-Lebesgue theorem, by which Götze had verified the inversion formula for Mehler-Fock integral transformation in [14].…”

“…Therefore the inversion formula is stated in the following. Theorem 1.9 ( [21]). If (X n , g) is of Q > 0 and of hypergeometric type, then any f ∈ C ∞,rad 0 (X) is recovered by…”

“…See [2,32]. Refer also to [21]. where PW(C) R even is the space of even, holomorphic functions h = h(λ) on C satisfying the following; for any N ∈ N there exists a constant c N > 0 such that…”