Sobolev algebra through a "carr\'e du champ" identity

TL;DR

This paper investigates the algebra properties of abstract Sobolev spaces linked to an operator, demonstrating that a 'carré du champ' identity extends these properties to a broader context.

Contribution

It establishes that the algebra property of Sobolev spaces holds under a 'carré du champ' identity, broadening previous results.

Findings

Algebra property holds under 'carré du champ' identity

Extends previous results to a wider class of Sobolev spaces

Provides a framework for further analysis of functional spaces

Abstract

We consider abstract Sobolev spaces of Bessel-type associated with an operator. In this work, we pursue the study of algebra properties of such functional spaces through the corresponding semigroup. As a follow-up of [4], we show that under the extra property of a "carr\'e du champ identity" , this algebra property holds in a wider range than previously shown.