  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.
