The composition operator in Sobolev Morrey spaces

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.