Fundamental solutions in the Colombeau framework: applications to solvability and regularity theory

In this article we introduce the notion of fundamental solution in the Colombeau context as an element of the dual ℒ(Gc(ℝ n), ℂ∼). After having proved the existence of a fundamental solution for a large class of partial differential operators with constant Colombeau coefficients, we investigate the relationships between fundamental solutions in ℒ(Gc(ℝn), ℂ∼), Colombeau solvability and G- and G∞ -hypoellipticity respectively. © 2008 Springer Science+Business Media B.V.