Complete characterization of functions which act, via superposition, on Sobolev spaces

Given a domain Í2 c RN and a Borel function h: /?„-»/?, conditions on h are sought ensuring that for every m-tuple of functions u¡ belonging to the first order Sobolev space W'*(Í2), the function A(w,(),. . ., «„(•)) will belong to a first order Sobolev space W '•'(Q), 1 < r < p < oo. In this paper conditions are found which are both necessary and sufficient in order that h have the above property. This result is based on a characterization obtained here for those Borel functions g: /lm X (RN)m -* R satisfying the requirement that for every m-tuple of functions u, e Wl*(Q,) the function g(ux(-)."„()> Vu,(-). . .., V«„,(•)) belongs to I/(Q). A needed result on the measurability of the set of Ä^-Lebesgue points of a function on RN is presented in an appendix.