Norm estimates and integral kernel estimates for a bounded operator in Sobolev spaces

Abstract

We show that a bounded linear operator from the Sobolev space $W^{-m}_{r}(\Omega)$ to $W^{m}_{r}(\Omega)$ is a bounded operator from $L_{p}(\Omega)$ to $L_{q}(\Omega)$, and estimate the operator norm, if $p,q,r\in [1,\infty]$ and a positive integer $m$ satisfy certain conditions, where $\Omega$ is a domain in $\mathbf{R}^{n}$. We also deal with a bounded linear operator from $W^{-m}_{p'}(\Omega)$ to $W^{m}_{p}(\Omega)$ with $p'=p/(p-1)$, which has a bounded and continuous integral kernel. The results for these operators are applied to strongly elliptic operators.