Real-variable theory and Fourier integral operators on semisimple Lie groups and symmetric spaces of real rank one

Let G be a non-compact connected semisimple Lie group of real rank one with finite center, K a maximal compact subgroup of G and X=G/K an associated symmetric space of real rank one. We will prove that L2,1G *L2,1 G⊆L 2,infinityG , which is a sharp endpoint estimate for the Kunze-Stein phenomenon. We will also show that the noncentered maximal operator M2fz =supz∈B 1B Bfz' dz'. is bounded from L2,1X to L2,infinityX and from LpX to LpX in the sharp range of exponents p∈&parl0;2,infinity&sqbr0; . The supremum in the definition of M2fz , is taken over all balls B containing the point z.; In the second part of this thesis we investigate Lp boundedness properties of a certain class of radial Fourier integral operators on the symmetric space X . We will prove that if ut is the solution at some fixed time t of the natural wave equation on X with initial data f and g and 1<p<infinity then ut LpX ≤Cp t fLp bpX +1+t g Lpbp-1 X . We will obtain both the precise behavior in t of the norm Cpt and the sharp regularity assumptions on the functions f and g (i.e. the exponent bp) that make this inequality possible. Our last theorem is concerned with the analog of Stein's maximal spherical averages introduced in [23] and we prove exponential decay estimates (of a highly non-Euclidean nature) on the Lp norm of supT≤t≤T+1 &vbm0;f*dstz &vbm0; .