Banach algebras of weakly differentiable functions

TL;DR

This paper establishes precise conditions under which Sobolev-type spaces built on rearrangement-invariant norms form Banach algebras, considering domain regularity and space parameters, with applications to Orlicz and Lorentz spaces.

Contribution

It provides a sharp balance condition linking Sobolev order, norm strength, and domain regularity for Banach algebra property, including new results for classical Sobolev spaces on irregular domains.

Findings

Characterization of Banach algebra conditions for Sobolev spaces

Analysis of domain regularity via isoperimetric functions

Results on boundedness of multiplication operators

Abstract

The question is addressed of when a Sobolev type space, built upon a general rearrangement-invariant norm, on an $n$-dimensional domain, is a Banach algebra under pointwise multiplication of functions. A sharp balance condition among the order of the Sobolev space, the strength of the norm, and the (ir)regularity of the domain is provided for the relevant Sobolev space to be a Banach algebra. The regularity of the domain is described in terms of its isoperimetric function. Related results on the boundedness of the multiplication operator into lower-order Sobolev type spaces are also established. The special cases of Orlicz-Sobolev and Lorentz-Sobolev spaces are discussed in detail. New results for classical Sobolev spaces on possibly irregular domains follow as well.