Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

Yoichi Miyazaki

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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.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 87, Number 10 (2011), 186-191.

First available in Project Euclid: 1 December 2011

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Zentralblatt MATH identifier

Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 35J40: Boundary value problems for higher-order elliptic equations

Sobolev space kernel theorem Sobolev embedding theorem elliptic operator


Miyazaki, Yoichi. Norm estimates and integral kernel estimates for a bounded operator in Sobolev spaces. Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 10, 186--191. doi:10.3792/pjaa.87.186.

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