Superposition Operators Between Higher-order Sobolev Spaces and a Multivariate Faà di Bruno Formula: Supercritical Case

This paper is a continuation of the work begun in [6] on superposition operators, (N u) (x) = g(u(x)), between two arbitrary Sobolev spaces. Sufficient conditions which ensure the well-definedness, the continuity and the validity of the higher-order chain rule for such operators are given in the supercritical case (see Remark 1.1). As a consequence of these properties, it is proved that N (W (Ω) ∩ W (Ω)) ⊂ W (Ω).