JournalszaaVol. 16, No. 3pp. 631–651

Differentiability Properties of the Autonomous Composition Operator in Sobolev Spaces

  • Massimo Lanza de Cristoforis

    Università di Padova, Italy
Differentiability Properties of the Autonomous Composition Operator in Sobolev Spaces cover
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Abstract

In this paper, we study the autonomous composition operator, which takes a pair of functions (f,g)(f,g) into its composite function f o gf\ o \ g. We assume that ff and gg belong to Sobolev spaces defined on open subsets of Rn\mathbb R^n, and we concentrate on the case in which the space for gg 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 ff and gg, the composition is continuous and differentiable with continuity up to order rr, with r1r ≥ 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