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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An embedding theorem


Authors: N. A. Chernyavskaya and L. A. Shuster
Journal: Proc. Amer. Math. Soc. 141 (2013), 3213-3221
MSC (2010): Primary 46E35; Secondary 34B24
Published electronically: June 3, 2013
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Abstract: We consider a weighted space $ W_1^{(2)}(\mathbb{R},q)$ of Sobolev type:

$\displaystyle W_1^{(2)}(\mathbb{R},q)=\left \{y\in AC_{\operatorname {loc}}^{(1... ... y''\Vert _{L_1(\mathbb{R})}+\Vert qy\Vert _{L_1(\mathbb{R})}<\infty \right \},$

where $ 0\le q\in L_1^{\operatorname {loc}}(\mathbb{R}) $ and

$\displaystyle \Vert y\Vert _{W_1^{(2)}(\mathbb{R},q)}=\Vert y''\Vert _{L_1(\mathbb{R})}+\Vert qy\Vert _{L_1(\mathbb{R})}.$

We obtain a precise condition which guarantees the embedding

$\displaystyle W_1^{(2)}(\mathbb{R},q)\hookrightarrow L_p(\mathbb{R}),\ p\in [1,\infty ).$


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Additional Information

N. A. Chernyavskaya
Affiliation: Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva, 84105, Israel

L. A. Shuster
Affiliation: Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel
Email: miriam@macs.biu.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11805-8
PII: S 0002-9939(2013)11805-8
Keywords: Embedding theorem
Received by editor(s): December 1, 2011
Published electronically: June 3, 2013
Communicated by: Michael Hitrik
Article copyright: © Copyright 2013 American Mathematical Society