Harmonic analysis for vector fields on hyperbolic spaces (Q1113416)

Vector fields on hyperbolic n-space \({\mathbb{H}}^ n\) are investigated with the methods of harmonic analysis. The algebra of G-invariant differential operators mapping vector fields to vector fields is determined (G the full isometry group of \({\mathbb{H}}^ n)\). Radial systems and spherical systems of vector fields are introduced. The spherical systems are expressed explicitly in terms of Jacobi functions. The spherical transform for radial systems of vector fields can be inverted. The Plancherel measure is calculated and a Paley-Wiener theorem is established. In analogy to Helgason's Fourier transform for functions on an arbitrary symmetric space, a Fourier transform for vector fields on \({\mathbb{H}}^ n\) is defined. The inversion formula for this transform is determined.