Multiplication in sobolev spaces, revisited

Abstract

In this artile, we re-examine some of the lassial pointwise multipliation theorems in Sobolev-Slobodekij spaes, in part motivated by a simple ounter-example that il-lustrates how ertain multipliation theorems fail in Sobolev-Slobodekij spaes when a bounded domain is replaed by ℝn. We identify the soure of the failure, and examine why the same failure is not enountered in Bessel potential spaes. To analyze the situation, we begin with a survey of the lassial multipliation results stated and proved in the 1977 artile of Zolesio, and arefully distinguish between the ase of spaes defined on the all of ℝn and spaes defined on a bounded domain (with e.g. a Lipshitz boundary). However, the survey we give has a few new wrinkles; the proofs we inlude are based almost exlusively on interpolation theory rather than Littlewood-Paley theory and Besov spaes, and some of the results we give and their proofs, inluding the results for negative exponents, do not appear in the literature in this form. We also inlude a partiularly important variation of one of the multipliation theorems that is relevant to the study of nonlinear PDE systems arising in general relativity and other areas. The onditions for mul-tipliation to be ontinuous in the ase of Sobolev-Slobodekij spaes are somewhat subtle and intertwined, and as a result, the multipliation theorems of Zolesio in 1977 have been ited (more than one) in the standard literature in slightly more generality than what is atually proved by Zolesio, and in ases that allow for onstrution of ounter-examples suh as the one inluded here.