1. INTRODUCTION

Let G = 5I(2,R), the group of real, 2 x 2 matrices of determinant one, K = SO(2),

the subgroup of rotations in the plane, and A the vector subgroup consisting of elements

which are positive on the main diagonal and zero elsewhere. For each 0 p ^ 2,

CP(G)

will denote the //-Schwartz space on G, defined in Section 3, and C (G) will denote the

zero-Schwartz space, defined as the intersection of all the //-Schwartz spaces.

The main result of this paper is the characterization of the images of Cp (G) and

C (G) under the operator-valued Fourier transform. The case p = 2 has a long history,

starting with Ehrenpreis and Mautner [11]. In their paper the image of a space similar

to C'(G) was characterized for G=PS£(2,R). Arthur, in [1], determined the image of

C (G) for any G of real rank one, while in [8,9] Eguchi considered the case where G has

just one conjugacy class of Cartan subgroups. The complete p = 2 result, for any reduc-

tive group, was established by Arthur in [2,3].

For values of p other than 2, Trombi [26] considered groups G of real rank one and

determined the image of CP(G:F), the subspace of CV(G) consisting of those functions

whose K-types fall within the finite set F. Kawazoe [20] considered the same problem,

but with a more restricted set of p values. Recently Trombi announced the solution of

the K-finite problem without the restriction to groups of real rank one. In our current

paper the restriction to finite K-type is removed for the group G = SL(2, R).

The basic organization of this paper comes from Trombi [26], with modifications to

remove the finite K-type restriction strongly influenced by the arguments in Arthur [1].

Much of the detailed structural information needed for SL(2, R) was derived from

material given by Milicic in [22]. The results concerning C (G) were developed in colla-

boration with Henrik Schlichtkrull.

Received by the editor March 23, 1987.

1