Tube Domains and the L 1 (Even L p ) HCS Spaces

Abstract

The HCS space of Chapter X corresponds exactly to the classical Schwartz space on iav under the spherical transform. When one restricts the spherical transform to various spaces intermediate to C c ∞ (K\G/K) and UCS(K\G/K) but ≠ nCS(K\G/K), then their spherical transforms extend beyond the imaginary axis iav of a C v , and the faster the functions on G decay, the larger is the domain of analytic continuation. The intermediate spaces can be defined by a very broad range of conditions, depending on applications. In this chapter we treat the L p -Schwartz spaces (0 < p < 2) originally considered by TrombiVaradarajan [TrV 71], whose treatment is essentially reproduced in [GaV 88], Chapter 7. Special cases in rank 1 had occurred before, e.g. in Trombi’s thesis (1970) and others, cf. the references given in [Ank 91]. Here we again follow Anker’s very simple and direct proofs by continuity from his version of the Paley-Wiener case [Ank 91]. Except for one item about tube domains, the proofs are the same as in Chapter X. The L1 case is especially important for the usual reasons that one needs L1 analysis, and also for our purposes in particular to deal with the heat kernel as in Chapter X, §7. Indeed, the conditions DIR 2 and DIR 3 are L1 conditions defining Dirac families.