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

摘要

This paper is a continuation of the work begun in [6] on superposition operators, (N_gu)(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_g (W~(m,p) (Ω) ∩ W_0~(k,p) (Ω)) ? W_0~(l,q)(Ω).