Microlocal Properties of Basic Operations in Colombeau Algebras

The Colombeau algebra of generalized functions allows us 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.