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December 2011 Norm estimates and integral kernel estimates for a bounded operator in Sobolev spaces
Yoichi Miyazaki
Proc. Japan Acad. Ser. A Math. Sci. 87(10): 186-191 (December 2011). DOI: 10.3792/pjaa.87.186

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.

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Yoichi Miyazaki. "Norm estimates and integral kernel estimates for a bounded operator in Sobolev spaces." Proc. Japan Acad. Ser. A Math. Sci. 87 (10) 186 - 191, December 2011. https://doi.org/10.3792/pjaa.87.186

Information

Published: December 2011
First available in Project Euclid: 1 December 2011

zbMATH: 1233.47031
MathSciNet: MR2863411
Digital Object Identifier: 10.3792/pjaa.87.186

Subjects:
Primary: 46E35
Secondary: 35J40

Keywords: elliptic operator , kernel theorem , Sobolev embedding theorem , Sobolev space

Rights: Copyright © 2011 The Japan Academy

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Vol.87 • No. 10 • December 2011
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