LP
HARMONIC ANALYSIS ON SL(2,R) 3
expect the most difficulties to occur when attempting to generalize to groups other than
SL(2, R).
The main results of Sections 12 through 14 are as follows. In Theorem 12.1 we
establish important bounds on the differences between the canonical matrix coefficients
and the j-th truncations of their Harish-Chandra expansions for both the principal and
discrete series representations. The most non-trivial and important aspect of the bounds
given here is their polynomial growth with respect to the K-type parameters n and m.
(Suppose n,m GZ. A function f on G is said to be spherical of type (nym) if it satisfies
equation (2.3). Any / E C (G) can be uniquely decomposed into a sum of spherical
functions / in C (G); the integers n and m which appear in this decomposition are
called the K-types of the function f.) The nature of the K-type dependence was not
needed, and hence was not considered, in [26].
In Section 13 the growth properties of the individual Harish-Chandra expansion
term coefficients are studied for the principal series. Theorem 13.7 gives the most impor-
tant bound. In this result an expansion term coefficient is multiplied by the Plancherel
measure density and a certain polynomial; A-derivatives of the result are shown to be
bounded by a polynomial in A and the K-types n and m. In Section 14 we handle the
discrete series by showing that their expansion coefficients come directly from those for
the principal series. This result, Theorem 14.3, ultimately rests on the Paley-Weiner
Theorem for SL(2, R) as established by Johnson [19].
Section 15 contains the key result of this paper, Theorem 15.2, a growth estimate on
fixed K-type components of the perturbated Lp wave packet portion of the inverse
transform 5 . For any subspace C C C (G), let Cm
n
denote those functions in C which
are spherical of type (m,n). If p is a defining seminorm on the space CP(G) and S ^'* is
that portion of the inverse transform which we conjecture takes ^H-mn(^)
*nto
C
H
.
mn
(G), then Theorem 15.2 gives that p{S^^f) is, loosely speaking, bounded by
specific derivatives of f, and is of polynomial growth in the K-types m and n. (If 7 =
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