Generalized Minkowski-funk Transforms and Small Denominators on the Sphere

The Cauchy problem for the Euler-Poisson-Darboux equation on the unit sphere Sn gives rise to a family of fractional integrals M cos f(x) which integrate f over the spherical cap of radius centered at the point x 2 Sn. These fractional integrals are called the generalized Minkowski-Funk transforms because various transforms of integral geometry (including those of Minkowski and Funk) are particular cases of M cos . The cases = 0; 1 and (1 n)=2 correspond to the spherical section transform, the spherical cap transform and the solution of the Cauchy problem for the wave equation on Sn respectively. Inversion of M cos with xed leads to the problem of small denominators which was not studied before in the context of the non-commutative harmonic analysis on spheres of dimension > 1. The structure of the kernel kerM cos and the behavior of M cos in Sobolev spaces are investigated depending on and arithmetical properties of . The paper sheds new light to the classical Schneider-Berenstein-Zalcman results on injectivity of the Pompeiu transform by giving them the relevant numbertheoretical meaning. Mathematics Subject Classi cation: Primary 42C10; Secondary 44A12, 35Q05, 52A30, 11D75