Functional properties of H\"ormander's space of distributions having a specified wavefront set

TL;DR

This paper investigates the topological and bornological properties of Hörmander's space of distributions with specified wavefront sets, revealing its nuclear, semi-reflexive, and complete normal space structure, with implications for quantum field theory.

Contribution

It provides a detailed analysis of the topological and bornological properties of Hörmander's distribution spaces, including dual space characteristics and convergence criteria, which were previously not fully understood.

Findings

$D'_$ is nuclear, semi-reflexive, and semi-Montel

Its dual $E'_$ is nuclear, barrelled, bornological, but not sequentially complete

Concrete criteria for membership, convergence, and boundedness in $D'_$

Abstract

The space $D'_\Gamma$ of distributions having their wavefront sets in a closed cone $\Gamma$ has become important in physics because of its role in the formulation of quantum field theory in curved space time. In this paper, the topological and bornological properties of $D'_\Gamma$ and its dual $E'_\Lambda$ are investigated. It is found that $D'_\Gamma$ is a nuclear, semi-reflexive and semi-Montel complete normal space of distributions. Its strong dual $E'_\Lambda$ is a nuclear, barrelled and bornological normal space of distributions which, however, is not even sequentially complete. Concrete rules are given to determine whether a distribution belongs to $D'_\Gamma$, whether a sequence converges in $D'_\Gamma$ and whether a set of distributions is bounded in $D'_\Gamma$.