The extension of distributions on manifolds, a microlocal approach

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.

arXiv commentThe main extension Theorem generalizes the result of the author's Phd thesis

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