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A note on composition operators in Sobolev spaces and an extension of the chain rule. (English) Zbl 0723.47027

Summary: We study the operator T defined by T[f,g](x)\(\equiv f(g(x))\) from special subsets of \(W^{m,p}(\Omega_ 1)\times (W^{m,q}(\Omega))^ n\) to \(W^{m,r}(\Omega)\). Conditions are found in order to ensure continuity and boundedness of T on bounded sets.

MSC:

47B38 Linear operators on function spaces (general)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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