Line measures and modular distributions

Abstract

Typically, operators with symbols of arithmetic interest, while acting from ${\mathcal{S}}(\mathbb{R})$ to ${\mathcal{S}^{\prime}}(\mathbb{R})$, have quite bad regularity properties. For instance, even before letting ${N} \nearrow {\infty}$, the ones considered in Chapter 4 could only act on continuous functions, yielding discrete measures as a result. Though building algebras of such operators would be of great interest in several questions, it is impossible to make sense, in general, of the composition of two such operators. In this chapter, we shall describe a way to turn around this difficulty, which has been successful in some instances.