Morrey type spaces and multiplication operator in Sobolev spaces

The paper deals with the operator $u\to gu$ defined in the Sobolev space $W^{r,p}(\mathrm{\Omega })$ and which takes values in $L^p(\mathrm{\Omega })$ when $\mathrm{\Omega }$ is an unbounded open subset in $R^n$. The functions $g$ belong to wider spaces of $L^p$ connected with the Morrey type spaces. $L^p$ estimates and compactness results are stated.