The inverse Abel transform for SU(p,q)
Let G/K be a Riemannian symmetric space of the noncompact type and \(G=KAN\) be an Iwasawa decomposition. The Abel transform is the mapping \(f\mapsto F_ f\) of \(C_ c^{\infty}(G//K)\) in \(C_ c^{\infty}(A)^ W\) (W is the Weyl group) which is defined by the formula \[ F_ f(a)=e^{\rho (\log a)}\int_{N}f(an) dn\quad. \] It is known that this mapping is an isomorphism (Gangolli). In the reviewed paper the inverse transform in the case \(G=SU(p,q)\) is described. It is a composition of some differential operator, a tensor product of fractional integrals (in number dim A) and multiplication with some fixed function. The proof is based on an explicit formula of Berezin and Karpelevich for the spherical functions. Moreover there is obtained an inversion formula for the Radon transform acting on K-invariant functions on the tangent space for G/K, \(G=SU(p,q)\). As an application of this latter formula a local regularity property for the corresponding Fourier transform is demonstrated.
scientific article; zbMATH DE number 3983837
scientific article