Banach algebras of weakly differentiable functions
Abstract
The question is addressed of when a Sobolev type space, built upon a general rearrangement-invariant norm, on an -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.
Keywords
Cite
@article{arxiv.1512.03198,
title = {Banach algebras of weakly differentiable functions},
author = {Andrea Cianchi and Luboš Pick and Lenka Slavíková},
journal= {arXiv preprint arXiv:1512.03198},
year = {2015}
}