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

DOI:
https://doi.org/10.1090/S0002-9939-97-03642-3

MathSciNet review:
1353406

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Abstract: We discuss spaces of Sobolev type which are defined by the operator with singularity: , where and . This operator appears in a one-dimensional harmonic oscillator governed by Wigner's commutation relations. We study smoothness of and continuity of () where 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:
https://doi.org/10.1090/S0002-9939-97-03642-3

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