LP HARMONIC ANALYSIS ON SL(2,R) 5
The tools needed to handle the integral case are developed in Section 17. Generaliz
ing on a method first introduced by Campoli [5], we construct a collection of functions
am
n
in the C^.mn(G) s P a c e s which have specified values at certain integer points.
(Actually the am
n
are constructed in Cg.m n(G) for some p' slightly smaller than p via
Theorem 16.3, the nonintegral isomorphism theorem.) Then, much as Trombi and
Ragozin did in [26], given any Kfinite F € Cg(G), we form an auxiliary function
/?p € Cg(G) such that the mapping F  F —F H(/?p) is continuous into
CH(G)
0

^n*s
mapping, the basic properties of which are summarized in Theorem 17.4, reduces the
proof of the isomorphism theorem in the integral case down to proving the following
statement: the mapping ^ ^S
R
F is continuous from the Kfinite elements in Cg(G)0
into
CP(G).
But this is just Corollary 15.6, a simple consequence of Theorem 15.2.
The results of Section 19 were developed in collaboration with Henrik Schlichtkrull.
In this section the zeroSchwartz space C (G) is introduced and its image under the
Fourier transform is characterized. This image is relatively easy to determine since
C (G) is the intersection of the //Schwartz spaces CP(G), 0 p 2, allowing us to
use the results developed earlier in the paper.
Most of the results of this paper were obtained while the author was a Visiting
Scholar at the University of Utah during the academic year 198283. I wish to thank the
University for their hospitality during that year. In particular I wish to thank Dragan
Miliecic' and Peter Trombi for many conversations and suggestions concerning this work.
They were both more than generous with their time and their help.
The results of Section 19 were obtained while the author was a visitor at the
Mathematical Sciences Research Institute in Berkeley during the academic year 198788.
MSRI is an ideal environment in which to work, and I wish to thank the staff for their
hospitality and support during the year.
In addition to his coauthoring of Section 19, I wish to thank Henrik Schlichtkrull
for carefully reading through the original manuscript and for suggesting significant