Quantitative functional calculus in Sobolev spaces

In the framework of Sobolev (Bessel potential) spaces H^n(R^d, R or C), we consider the nonlinear Nemytskij operator sending a function x in R^d ->f(x) into a composite function x in R^d -> 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 in a fully quantitative way the H^n norm of the function x ->G(f(x),x). When applied to the function 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.