On the foundations of nonlinear generalized functions II

Abstract

This paper gives a comprehensive analysis of algebras of Colombeau-type generalized functions in the range between the diffeomorphism-invariant quotient algebra ${\mathcal{G}}^d={\mathcal{E}}_M/\mathcal{N}$ introduced in part I and Colombeau's original algebra ${\mathcal{G}}^e$. Three main results are established: First, a simple criterion describing membership in $\mathcal{N}$ (applicable to all types of Colombeau algebras) is given. Second, two counterexamples demonstrate that ${\mathcal{G}}^d$ is not injectively included in ${\mathcal{G}}^e$. Finally, it is shown that in the range ``between'' ${\mathcal{G}}^d$ and ${\mathcal{G}}^e$ only one more construction leads to a diffeomorphism invariant algebra. In analyzing the latter, several classification results essential for obtaining an intrinsic description of ${\mathcal{G}}^d$ on manifolds are derived.

43 pages, LaTeX