Spingroups and spherical means. III. (English) Zbl 0709.35077

Proc. Winter Sch. Geom. Phys., SrnĂ­/Czech. 1988, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 21, 295-323 (1989).
[For the entire collection see Zbl 0672.00006.]
[For part II see Suppl. Rend. Circ. Mat. Palermo, II. Ser. 14, 157-177 (1987; Zbl 0651.35073).]
Author’s summary: In this paper we study the mean values of functions defined on the unit sphere \(S^{m-1}\) over spheres of a given dimension inside \(S^{m-1}\). These mean value operators satisfy certain first order systems of differential equations with values in a Clifford algebra, which are generalizations of the Darboux equation. We construct explicit solutions to these equations and apply the solution to the Radon transforms on the unit sphere, leading to an improvement of the inversion formula obtained by S. Helgason. We also show that the Radon transform over geodesic spheres of dimension p-1 may be expressed in terms of the Radon transform over spheres of codimension p inside \(S^{m-1}\).
Reviewer: J.Wloka


35Q05 Euler-Poisson-Darboux equations
35A22 Transform methods (e.g., integral transforms) applied to PDEs