A Higher-level Weyl Calculus of Operators

Abstract

The Opt-calculus, to be introduced in Section9 with a first approach in the present section, smoothes up the most serious difficulties inherent in the automorphic pseudodifferential analysis in two ways. First, as soon as p > 1, it “forgets” all distributions homogeneous of degree-1: and a detailed spectral analysis of our problem shows that this part of the decomposition is indeed the major obstacle to defining the sharp product of, say, two Eisenstein distributions. Next, its use makes it possible to substitute for the collection of functions (2p _z ) a set of functions z+i which are just as nice as the IN ‘s outside zero but which, moreover, vanish up to a certain order at zero (cf. (6.2)). In the next seven sections, we shall develop this calculus to a further extent than what is strictly needed for its application to the automorphic pseudodifferential calculus. The horocyclic calculus was first introduced in [56, Theorem 6.1], but [62, Section17] gives a more self-contained introduction.