Microlocal properties of basic operations in Colombeau algebras

Abstract

The Colombeau algebra of generalized functions allows to unrestrictedly carry out products of distributions. We analyze this operation from a microlocal point of view, deriving a general inclusion relation for wave front sets of products in the algebra. Furthermore, we give explicit examples showing that the given result is optimal, i.e. its assumptions cannot be weakened. Finally, we discuss the interrelation of these results with the concept of pullback under smooth maps.