The Composition Operator in Sobolev Morrey Spaces

Abstract

In this paper we prove sufficent conditions on a map \( f \) from the real line to itself in order that the composite map \( f \circ g \) belongs to a Sobolev Morrey space of real valued functions on a domain of the \( n \)-dimensional space for all functions \( g \) in such a space. Then we prove sufficient conditions on \( f \) in order that the composition operator \( T_f \) defined by \( T_f[g] \equiv f \circ g \) for all functions \( g \) in the Sobolev Morrey space is continuous, Lipschitz continuous and differentiable in the Sobolev Morrey space. We confine the attention to Sobolev Morrey spaces of order up to one.