Quantitative functional calculus in Sobolev spaces

In the frame work of Sobolev (Bessel potential) spaces Hn(Rd,R or C), we consider the nonlinear Nemytskij operator sending a function x∈Rd↦f(x) into a composite function x∈Rd↦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↦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.