Embedding and compactness results for multiplication operators in Sobolev spaces

The paper deals with the multiplication operator gu defined in the Sobolev space $W^{r,p}(\Omega)$ and which takes values in $L^{p}$ $(\Omega)$ when $\Omega$ is an unbounded open subset in $R^n$. The functions g belong to Morrey type spaces which provide an intermediate space between $L^{\infty}$ and $L^{p}_{loc}(\Omega)$. The main result is a embedding result from which we can deduce a Fefferman type inequality. $L^{p}$ estimates and a compactness result are also stated.