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

Abstract:

Given a domain $\Omega \subset {R_N}$ and a Borel function $h: {R_m} \to R$, conditions on h are sought ensuring that for every m-tuple of functions ${u_i}$ belonging to the first order Sobolev space ${W^{1,p}}(\Omega )$, the function $h({u_1}( \cdot ), \ldots ,{u_m}( \cdot ))$ will belong to a first order Sobolev space ${W^{1,r}}(\Omega )$, $1 \leqslant r \leqslant p < \infty$.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 $g: {R_m} \times {({R_N})_m} \to R$ satisfying the requirement that for every m-tuple of functions ${u_i} \in {W^{1,p}}(\Omega )$ the function $g({u_1}( \cdot ), \ldots ,{u_m}( \cdot ),\nabla {u_1}( \cdot ), \ldots ,\nabla {u_m}( \cdot ))$ belongs to ${L^r}(\Omega )$. A needed result on the measurability of the set of ${R_k}$-Lebesgue points of a function on ${R_N}$ is presented in an appendix.