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Journal of Function Spaces and Applications
Volume 2 (2004), Issue 3, Pages 279-321
http://dx.doi.org/10.1155/2004/832750

Quantitative functional calculus in Sobolev spaces

1Dipartimento di Matematica, Politecnico di Milano, P.za L. da Vinci 32, I-20133 Milano, Italy
2Dipartimento di Matematica, Università di Milano, Via C. Saldini 50, I-20133 Milano, Italy
3Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italy

Received 1 June 2003

Academic Editor: Jürgen Appell

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the frame work of Sobolev (Bessel potential) spaces Hn(Rd,RorC), we consider the nonlinear Nemytskij operator sending a function xRdf(x) into a composite function xRdG(f(x),x). Assuming sufficient smoothness for G, we give a “tame” bound on the Hn norm of this composite function in terms of a linear function of the Hn norm of f, with a coefficient depending on G and on the Ha norm of f, for all integers n,a,d with a>d/2. In comparison with previous results on this subject, our bound is fully explicit, allowing to estimate quantitatively the Hn norm of the function xG(f(x),x). When applied to the case G(f(x),x)=f2(x), this bound agrees with a previous result of ours on the pointwise product of functions in Sobolev spaces.