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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 of Sobolev type
for an operator with singularity


Author: Shuji Watanabe
Journal: Proc. Amer. Math. Soc. 125 (1997), 839-848
MSC (1991): Primary 35G10, 46E35, 47B25
MathSciNet review: 1353406
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Abstract: We discuss spaces of Sobolev type which are defined by the operator with singularity: ${\cal D} = d/dx - (c/x)R$, where $Ru(x) = u(-x)$ and $c > 1$. This operator appears in a one-dimensional harmonic oscillator governed by Wigner's commutation relations. We study smoothness of $u$ and continuity of $u / x^{\beta }$ ($\beta > 0$) where $u$ is in each space of Sobolev type, and obtain a generalization of the Sobolev embedding theorem. On the basis of a generalization of the Fourier transform, the proof is carried out. We apply the result to the Cauchy problems for partial differential equations with singular coefficients.


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Additional Information

Shuji Watanabe
Affiliation: Department of Mathematics, Toyota National College of Technology, Eisei-cho 2-1, Toyota-shi 471, Japan

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03642-3
PII: S 0002-9939(97)03642-3
Keywords: Embedding theorem of Sobolev type, operator with singularity, partial differential equations with singular coefficients
Received by editor(s): September 22, 1995
Additional Notes: Research partially supported by Grant-in-Aid for Scientific Research (No. 07740175), Ministry of Education, Science, Sports and Culture
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society