Localization properties of highly singular generalized functions

TL;DR

This paper investigates the localization characteristics of highly singular generalized functions within broad spaces of entire analytic test functions, introducing a notion of carrier cone to replace support for these functionals.

Contribution

It defines and establishes the properties of carrier cones for analytic functionals, including their uniqueness and minimality, in the context of highly singular generalized functions.

Findings

Carrier cones can be correctly defined for these functionals.

Each functional has a uniquely determined minimal carrier cone.

The framework applies to Gelfand--Shilov spaces with eta<1.

Abstract

We study the localization properties of generalized functions defined on a broad class of spaces of entire analytic test functions. This class, which includes all Gelfand--Shilov spaces $S^\beta_\alpha(\R^k)$ with $\beta<1$, provides a convenient language for describing quantum fields with a highly singular infrared behavior. We show that the carrier cone notion, which replaces the support notion, can be correctly defined for the considered analytic functionals. In particular, we prove that each functional has a uniquely determined minimal carrier cone.