The extension of distributions on manifolds, a microlocal approach

TL;DR

This paper develops microlocal conditions for extending distributions on manifolds with submanifolds, with applications to quantum field theory renormalization on curved spacetimes.

Contribution

It introduces geometric scaling conditions and microlocal criteria for distribution extension, generalizing previous results and optimizing wave front set control.

Findings

Established conditions for distribution extension using geometric scaling.

Provided microlocal criteria to control wave front sets of extensions.

Identified a subspace with minimal wave front set extensions applicable to quantum field theory.

Abstract

Let $M$ be a smooth manifold, $I\subset M$ a closed embedded submanifold of $M$ and $U$ an open subset of $M$. In this paper, we find conditions using a geometric notion of scaling for $t\in \mathcal{D}^\prime(U\setminus I)$ to admit an extension in $\mathcal{D}^\prime(U)$. We give microlocal conditions on $t$ which allow to control the wave front set of the extension generalizing a previous result of Brunetti--Fredenhagen. Furthermore, we show that there is a subspace of extendible distributions for which the wave front of the extension is minimal which has applications for the renormalization of quantum field theory on curved spacetimes.