Quantitative functional calculus in Sobolev spaces

摘要

In the frame work of Sobolev (Bessel potential) spacesHn(Rd,R?or?C), we consider the nonlinear Nemytskij operator sending a functionx∈Rd?f(x)into a composite functionx∈Rd?G(f(x),x). Assuming sufficient smoothness forG, we give a “tame” bound on theHnnorm of this composite function in terms of a linear function of theHnnorm off, with a coefficient depending onGand on theHanorm off, for all integersn,a,dwitha>d/2. In comparison with previous results on this subject, our bound is fully explicit, allowing to estimate quantitatively theHnnorm of the functionx?G(f(x),x). When applied to the caseG(f(x),x)=f2(x), this bound agrees with a previous result of ours on the pointwise product of functions in Sobolev spaces.