Note on the Taylor expansion of smooth functions defined on Sobolev spaces (Q1177323)

Let \(H^ \sigma(\mathbb{R}^ n)\) be the Sobolev space (called often \(W^{\sigma,2} (\mathbb{R}^ n)\)), \(u\) a real-valued function in this space, and \(f\) a rapidly decreasing function on \(\mathbb{R}\). The composition function \(f(u)\) can be considered as a function from \(H^ \sigma(\mathbb{R}^ n)\) and we have the Taylor expansion for \(f\) of the form \[ f(v+u)= \sum_{k=0}^{m-1} {{u^ k} \over {k!}} f^{(k)}(v)+ R_ m(f)(v,u), \] where \(u,v\in H^ \sigma (\mathbb{R}^ n)\) and \(R_ m\) is the \(m\)th remainder. In the paper the estimate is given for the norm of \(R_ m\) in the space \(H^ s(\mathbb{R}^ n)\), where \(0\leq s\leq\sigma\) and \(\sigma> {n\over 2}+1\) or \(\sigma>{n\over 2}\) and \(\sigma\geq 1\), an extension of the result of \textit{J. Rauch} and \textit{M. C. Reed} [Duke Math. J. 49, 397-475 (1972; Zbl 0503.35055)].