In this paper, we study the autonomous composition operator, which takes a pair of functions into its composite function . We assume that and belong to Sobolev spaces defined on open subsets of , and we concentrate on the case in which the space for 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 and , the composition is continuous and differentiable with continuity up to order , with . 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. 631–651DOI 10.4171/ZAA/782