Multiplication in Sobolev spaces, revisited

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