On superposition operators between higher-order Sobolev spaces and a multivariate Faà di Bruno formula: the subcritical case

Abstract

In this paper, superposition operators, $ (N_gu ) (x )= g (u (x ) )$, between two arbitrary Sobolev spaces are studied. 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 subcritical case (see Remark 1.1). As a consequence of these properties, it is proved that $N_g (W^{m,p} (\Omega )\cap W_0^{k,p} (\Omega ) )\subset W_0^{l,q} (\Omega )$.