Quantitative functional calculus in Sobolev spaces

In the framework of Sobolev (Bessel potential) spaces Hn(R,R or C), we consider the nonlinear Nemytskij operator sending a function x ∈ R 7→ f(x) into a composite function x ∈ R 7→ G(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 x 7→ G(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.