2 WILLIAM H. BARKER We now describe the contents of the various sections of this paper. Sections 2 and 3 introduce general notation and the Lp- Schwartz spaces CP(G). Sections 4 through 8 consider the basic properties of the canonical matrix coefficients for both the principal series and the discrete series of representations of SL(2, R). Most of the results in these sections are already known, although locating some of them in the literature can be time consuming. The major result of these sections is Theorem 8.1, a slight extension for SL(2,R) of the discrete series matrix coefficient bound of Trombi and Varadarajan [27]. In Sections 9 through 11 we return to CP(G), considering the basic properties of the Fourier transform T on this space and the natural decomposition of CP(G) into the Lp cusp forms space Cg(G) and the perturbated Lp wave packets space Cg(G). The space CP(G) = C*(G) x C^G), which we will show to be the image of C p (G) under T , is defined in Section 9, and T is shown to be a continuous injection of CP(G) into CP(G) (Theorem 9.6). From Arthur's work [1] it is known that T is an isomorphism from C (G) onto C (G), with the inverse denoted by S. Establishing the surjectivity of T from CP(G) onto CP(G), and the continuity of S on CP(G), are the goals of the remaining portions of the paper. Sections 12 through 14 undertake a detailed study of the Harish-Chandra expan- sions for matrix coefficients of the principal series and discrete series for SL(2, R). To determine the image of C (G) under the Fourier transform for any reductive group G, only the first, or "constant", term needs to be considered. This remains true for CP(G) when G = SL(2, R) and 1 p 2. However, for p 1, we need to make heavy use of further terms in the Harish-Chandra expansions, the number of required terms increasing as p decreases toward zero. Truncating the expansions at the required number of terms yields asymptotic approximations to the matrix coefficients on the vector group A, and it is these approximations that will be needed in the subsequent sections. The proofs of all the major results in these sections depend on being able to explicitly write out the differential equations that the matrix coefficients must satisfy. This is where we would
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