The Extension of Distributions on Manifolds,a Microlocal Approach

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.