Regularity of nonlinear generalized functions: a counterexample in the nonstandard setting

TL;DR

This paper demonstrates that a key regularity property in Colombeau-type generalized function algebras fails in the nonstandard setting, challenging assumptions about regularity transfer from classical to nonstandard frameworks.

Contribution

It provides a counterexample showing the breakdown of ${ m extbf{G}}^ extbf{ extit{ ext{in}}}$-regularity in nonstandard Colombeau algebras, highlighting a fundamental difference from the standard theory.

Findings

${ m extbf{G}}^ extbf{ extit{ ext{in}}}$-regularity does not hold in nonstandard Colombeau algebras

Counterexample illustrating failure of regularity property

Challenges assumptions about regularity in nonstandard generalized functions

Abstract

Regularity theory in generalized function algebras of Colombeau type is largely based on the notion of ${\mathcal G}^\infty$-regularity, which reduces to $C^\infty$-regularity when restricted to Schwartz distributions. Surprisingly, in the nonstandard version of the Colombeau algebras, this basic property of ${\mathcal G}^\infty$-regularity does not hold.