Quantitative functional calculus in Sobolev spaces

In the framework of Sobolev (Bessel potential) spaces $H^n(\reali^d, \reali {or} \complessi)$, we consider the nonlinear Nemytskij operator sending a function $x\in {\text{reali}}^d\mapsto f(x)$ into a composite function $x \in \reali^d \mapsto G(f(x), x)$\mathrm{.}Assumingsufficientsmoothnessfor$G$, 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 $a>d\mathrm{/}2$. 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.