  In this article, we re-examine some of the classical pointwise multiplication
theorems in Sobolev-Slobodeckij spaces, in part motivated by a simple
counter-example that illustrates how certain multiplication theorems fail in
Sobolev-Slobodeckij spaces when a bounded domain is replaced by Rn. We identify
the source of the failure, and examine why the same failure is not encountered
in Bessel potential spaces. To analyze the situation, we begin with a survey of
the classical multiplication results stated and proved in the 1977 article of
Zolesio, and carefully distinguish between the case of spaces defined on the
all of Rn and spaces defined on a bounded domain (with e.g. a Lipschitz
boundary). However, the survey we give has a few new wrinkles; the proofs we
include are based almost exclusively on interpolation theory rather than
Littlewood-Paley theory and Besov spaces, and some of the results we give and
their proofs, including the results for negative exponents, do not appear in
the literature in this form. We also include a particularly important variation
of one of the multiplication theorems that is relevant to the study of
nonlinear PDE systems arising in general relativity and other areas. The
conditions for multiplication to be continuous in the case of
Sobolev-Slobodeckij spaces are somewhat subtle and intertwined, and as a
result, the multiplication theorems of Zolesio in 1977 have been cited (more
than once) in the standard literature in slightly more generality than what is
actually proved by Zolesio, and in cases that allow for the construction of
counter-examples such as the one included here.
