# Differentiability Properties of the Autonomous Composition Operator in Sobolev Spaces

### Massimo Lanza de Cristoforis

Università di Padova, Italy

## Abstract

In this paper, we study the autonomous composition operator, which takes a pair of functions $(f,g)$ into its composite function $f\ o \ g$. 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 $r ≥ 1$. Then we show the optimality of such conditions by means of theorems of ’inverse’ type.

## Cite this article

Massimo Lanza de Cristoforis, Differentiability Properties of the Autonomous Composition Operator in Sobolev Spaces. Z. Anal. Anwend. 16 (1997), no. 3, pp. 631–651

DOI 10.4171/ZAA/782