Morrey type spaces and multiplication operator in Sobolev spaces

TL;DR

This paper investigates the properties of multiplication operators in Sobolev spaces, focusing on functions in Morrey type spaces, and establishes $L^p$ estimates and compactness results for unbounded domains.

Contribution

It introduces new conditions on functions in Morrey type spaces that ensure boundedness and compactness of multiplication operators in Sobolev spaces.

Findings

Established $L^p$ estimates for the multiplication operator.

Proved compactness results for the operator in unbounded domains.

Extended the class of functions for which the operator is bounded.

Abstract

The paper deals with the operator $u\rightarrow gu$ 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 wider spaces of $L^p$ connected with the Morrey type spaces. $L^p$ estimates and compactness results are stated.