L1Estimates for Maximal Functions and Riesz Transform on Real Rank 1 Semisimple Lie Groups

Fundamental domains for lattices in rank one semisimple lie groups.

We construct a fundamental domain omega for an arbitrary lattice [unk] in a real rank one, real simple Lie group, where omega has finitely many cusps (i.e., is a finite union of Siegel sets) and has the Siegel property (i.e., the set {gamma [unk] [unk]|omegagamma [unk] omega [unk] varphi} is finite). From the existence of omega we derive a number of consequences. In particular, we show that [un...

Harmonic Analysis of Tempered Distributions on Semisimple Lie Groups of Real Rank One

Let G be a real semisimple Lie group. Harish-Chandra has defined the Schwartz space, V[G), on G. A tempered distribution on G is a continuous linear functional on R G ) . If the real rank of G equals one, Harish-Chandra has published a version of the Plancherel formula for I^(G) [3(k), 5241. We restrict the Fourier transform map to %(G), and we compute the image of the space V(G) [Theorem 31. T...

Orbit Functions of Compact Semisimple Lie Groups as Special Functions

An orbit function is the contribution to an irreducible character of a compact semisimple Lie group G of rank n from one of its Weyl group orbits. Properties of such functions will be described for compact simple Lie groups of all types. In particular, products of the functions decompose into their sums, the functions are periodic on copies of the fundamental region F of G, they are solutions o...

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 ...

Equivariant derived category and representation of real semisimple Lie groups