Fourier integral operators on noncompact symmetric spaces of real rank one

Let X = G/K be a noncompact symmetric space of real rank one. The purpose of this paper is to investigate L-P boundedness properties of a certain class of radial Fourier integral operators on the space X. We will prove that if u(tau) is the solution at some ilxed time tau of the natural wave equation on X with initial data f and g and 1 < p < infinity, then parallel to u(tau)parallel to(Lp(X)) less than or equal to C-p(tau)(parallel to f parallel to(Lhpp(X)) + (1+tau) parallel to g parallel to L-hpp, ((X))). We will obtain both the precise behavior of the norm C-p(tau) and the sharp regularity assumptions on the functions f and g (i.e.. the exponent b(p)) that make this inequality possible. in the second part of the paper we deal with the analog of E. M, Stein's maximal spherical averages and prove exponential decay estimates (of a highly non-euclidean nature) on the L-P norm of sup(T less than or equal to tau less than or equal to T.1) \f*d sigma(tau)(z)\, where d sigma(tau) is a normalized spherical measure. (C) 2000 Academic Press.