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 L(G) ∗ L(G) ⊆ L(G), which is a sharp endpoint estimate for the Kunze-Stein phenomenon. We will also show that the noncentered maximal operator M2f(z) = sup z∈B 1 |B| ∫ B f(z)dz is bounded from L(X) to L(X) and from L(X) to L(X) in the sharp range of exponents p ∈ (2,∞]. The supremum in the definition of M2f(z), is taken over all balls B containing the point z. In the second part of this thesis we investigate L boundedness properties of a certain class of radial Fourier integral operators on the symmetric 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 Lbp (X) + (1 + Lbp−1(X) ) . We will obtain both the precise behavior in τ 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. 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 L norm of sup T≤τ≤T+1 |f ∗ dστ (z)|.