Strong ultra-regularity properties for positive elements in the twisted convolutions

TL;DR

This paper proves that positive elements in twisted convolutions with ultra-test function properties at the origin maintain these properties globally, with applications to positive semi-definite Weyl operators.

Contribution

It establishes a global ultra-regularity property for positive elements in twisted convolutions, extending local properties to the entire domain.

Findings

Positive elements with local ultra-test properties are globally ultra-test.

Application to positive semi-definite Weyl operators.

Extension of local regularity to global regularity in twisted convolutions.

Abstract

We show that positive elements with respect to the twisted convolutions, belonging to some ultra-test function space of certain order at origin, belong to the ultra-test function space of the same order everywhere. We apply the result to positive semi-definite Weyl operators.