Differentiability properties of the autonomous composition operator in Sobolev spaces
The author studies the autonomous composition operator, which takes a pair of functions \((f,g)\) into its composite function \(f\circ g\). He assumes that \(f\) and \(g\) belong to Sobolev spaces defined on open subsets of \(\mathbb{R}^n\), and he concentrates on the case in which the space for \(g\) is a Banach algebra. He gives a sufficient condition in order that the composition maps bounded sets to bounded sets, and exploits the density of the polynomial functions in the space for \(f\) in order to prove that for suitable Sobolev exponents of the spaces for \(f\) and \(g\), the composition is continuous and differentiable with continuity up to order \(r\), with \(r\geq 1\). Then he shows the optimality of such conditions by means of theorems of `inverse' type.
scientific article; zbMATH DE number 1068014
scientific article