Suppleness of the sheaf of algebras of generalized functions on manifolds

TL;DR

This paper demonstrates that certain sheaves of generalized functions on manifolds are supple, highlighting a key difference from the non-suppleness of the sheaf of distributions, with implications for the structure of these function spaces.

Contribution

It establishes the suppleness of sheaves of generalized functions on manifolds, contrasting with the non-suppleness of the distribution sheaf, advancing understanding of their algebraic properties.

Findings

Sheaves of generalized functions are supple.

Distribution sheaves are not supple.

Contrasts between generalized functions and distributions are clarified.

Abstract

We show that the sheaves of algebras of generalized functions $\Omega\to \mathcal{G}(\Omega)$ and $\Omega\to \mathcal{G}^{\infty}(\Omega)$, $\Omega$ are open sets in a manifold $X$, are supple, contrary to the non-suppleness of the sheaf of distributions.