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 Lp boundedness properties of a certain class of radial Fourier integral operators on the space X. We will prove that if uτ is the solution at some fixed time τ of the natural wave equation on X with initial data f and g and 1<p<∞, then ‖uτ‖Lp(X)⩽Cp(τ)(‖f‖Lpbp(X)+(1+τ) ‖g‖Lpbp−1(X)). We will obtain both the precise behavior of the norm Cp(τ) and the sharp regularity assumptions on the functions f and g (i.e., the exponent bp) that make this inequality possible. In the second part of the paper we deal with the analog of E. M. Steinʹs maximal spherical averages and prove exponential decay estimates (of a highly non-euclidean nature) on the Lp norm of supT⩽τ⩽T+1 |f*dστ(z)|, where dστ is a normalized spherical measure.