The equality of the homogeneous and the Gabor wave front set

TL;DR

This paper proves the equivalence of two wave front set concepts for tempered distributions, constructs distributions with specific wave front sets, and develops a pseudodifferential calculus tailored to Nakamura's homogeneous wave front set.

Contribution

It establishes the equality of H"ormander's and Nakamura's wave front sets, introduces a method to construct distributions with prescribed wave front sets, and develops a specialized pseudodifferential calculus.

Findings

H"ormander's and Nakamura's wave front sets coincide.

A construction method for distributions with a given wave front set.

A pseudodifferential calculus adapted to Nakamura's wave front set.

Abstract

We prove that H\"ormander's global wave front set and Nakamura's homogeneous wave front set of a tempered distribution coincide. In addition we construct a tempered distribution with a given wave front set, and we develop a pseudodifferential calculus adapted to Nakamura's homogeneous wave front set.