Journal of Integral Equations and Applications

The compactness of a weakly singular integral operator on weighted Sobolev spaces

David Elliott

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Abstract

It is shown that the weakly singular integral operator~$\int_{-1}^{1}\big(\phi(\tau)/|\tau -t|^{\gamma}\big)\,d\tau$, where $0\lt \gamma\lt 1$, maps the weighted Sobolev space~$W_{p;\alpha,\beta}^{(n)}(\Omega)$ compactly into itself for $1\lt p\lt \infty$, $0\lt \alpha+1/q, \beta+1/q\lt 1$~and $n\in \mathbb{N}_0$.

Article information

Source
J. Integral Equations Applications, Volume 26, Number 4 (2014), 483-496.

Dates
First available in Project Euclid: 9 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1420812882

Digital Object Identifier
doi:10.1216/JIE-2014-26-4-483

Mathematical Reviews number (MathSciNet)
MR3299828

Zentralblatt MATH identifier
1323.46026

Subjects
Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

Citation

Elliott, David. The compactness of a weakly singular integral operator on weighted Sobolev spaces. J. Integral Equations Applications 26 (2014), no. 4, 483--496. doi:10.1216/JIE-2014-26-4-483. https://projecteuclid.org/euclid.jiea/1420812882


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References

  • David Elliott and Susumu Okada, The finite Hilbert transform and weighted Sobolev spaces, Math. Nachr. 266 (2004), 34–47.
  • A. Kufner, Weighted Sobolev spaces, Wiley & Sons, New York, 1985.
  • S.G. Mikhlin and S. Prössdorf, Singular integral operators, Springer-Verlag, Berlin, 1986.
  • F.W.J. Olver, et al., NIST Handbook of mathematical functions, NIST and Cambridge University Press, New York, 2010.