 
Summary: Harmonic Analysis of Tempered Distributions
on Semisimple Lie Groups
of Real Rank One
JAMES G. ARTHUR
SUMMARY.Let G be a real semisimple Lie group. HarishChandra 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, HarishChandra 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. This permits us to develop the theory of harmonic analysis
for tempered distributions on G [Theorem 51.
Summary
1.Introduction
2. Preliminaries
3. Plancherel formula for L2(G)
4. Statement of Theorem 3
5. Spherical functions
6. Proof that the map is injective
7. Theorem 3' and some elementary formulae
