Differentiability Properties of the Autonomous Composition Operator in Sobolev Spaces

Abstract

In this paper, we study the autonomous composition operator, which takes a pair of functions $(f,g)$ into its composite function $f\circ g$. We assume that $f$ and $g$ belong to Sobolev spaces defined on open subsets of ${\mathbb{R}}^n$, and we concentrate on the case in which the space for $g$ is a Banach algebra. We give a sufficient condition in order that the composition maps bounded sets to bounded sets, and we exploit the density of the polynomial functions in the space for I 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 $r\ge 1$. Then we show the optimality of such conditions by means of theorems of ‘inverse’ type.