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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An embedding theorem of Sobolev type for an operator with singularity
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by Shuji Watanabe PDF
Proc. Amer. Math. Soc. 125 (1997), 839-848 Request permission

Abstract:

We discuss spaces of Sobolev type which are defined by the operator with singularity: $\mathcal {D} = d/dx - (c/x)R$, 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
  • 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
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 839-848
  • MSC (1991): Primary 35G10, 46E35, 47B25
  • DOI: https://doi.org/10.1090/S0002-9939-97-03642-3
  • MathSciNet review: 1353406