Rapidly converging approximations and regularity theory

Abstract

We consider distributions on a closed compact manifold \(M\) as maps on smoothing operators. Thus spaces of maps between \({{\Psi }^{\!-\!\infty }}(M)\) and \(\mathcal{C ^{\infty }}(M)\) are considered as generalized functions. For any collection of regularizing processes we produce various algebras of generalized functions and equivariant embeddings of distributions into such algebras. The regularity for such generalized functions is provided in terms of a certain tameness of maps between graded Frechét spaces. This also recovers the singularity behaviour of distributions (singular support/wavefront sets) in terms of certain subalgebras of the algebra of generalized functions. This notion of regularity is compared with the regularity in Colombeau algebras in the \(\mathcal{G }^{\infty }\) sense.