Quantitative functional calculus in Sobolev spaces

Details Quantitative functional calculus in Sobolev spaces Quantitative functional calculus in Sobolev spaces

2003-05-23

arxiv.org/abs/math/0305331v3

Journal of Function Spaces and Applications (JFSA) 2 (2004), 279-321

Mathematics

Functional Analysis

LaTex, 37 pages. Final version, differing only by minor typographical changes from the versions of May 23, 2003 and March 8, 2

Scientific paper

In the framework of Sobolev (Bessel potential) spaces $H^n(\reali^d, \reali {or} \complessi)$, we consider the nonlinear Nemytskij operator sending a function $x \in \reali^d \mapsto f(x)$ into a composite function $x \in \reali^d \mapsto G(f(x), x)$. Assuming sufficient smoothness for $G$, we give a "tame" bound on the $H^n$ norm of this composite function in terms of a linear function of the $H^n$ norm of $f$, with a coefficient depending on $G$ and on the $H^a$ 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 $H^n$ norm of the function $x \mapsto G(f(x),x)$. When applied to the case $G(f(x), x) = f^2(x)$, this bound agrees with a previous result of ours on the pointwise product of functions in Sobolev spaces.

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