ALMOST EVERYWHERE CONVERGENCE OF INVERSE SPHERICAL TRANSFORMS ON NONCOMPACT SYMMETRICAL SPACES

Let G be a connected noncompact semisimple Lie group with finite cente r and real rank one. Fix a maximal subgroup K. We consider K bi-invari ant Functions f an G and their spherical transform (f) over cap(lambda ) = integral(G) f(g) <(phi)over bar>(lambda)(g) dg, where phi(lambda) denote the elementary spherical functions on Gill and lambda greater t han or equal to 0 . We consider the maximal operators S f(t) = Sup(R> 1)/ integral(1)(R) (f) over cap(lambda)phi(lambda)(alpha(t))/c(lambda) /(-2)d lambda/ and prove that S maps boundedly L-K(S)K(G) --> L-s(G) + L-2(G) for 2n/(n+1) < s less than or equal to 2 where n = dim(G/K). The result is sharp and it implies a.e. convergence properties of the inverse spherical transforms. (C) 1997 Academic Press.