Skip to main content
SpringerLink
Log in

An Algebra of Shift-invariant Singular Integral Operators with Slowly Oscillating Data and Its Application to Operators with a Carleman Shift

  • Conference paper
Analysis, Partial Differential Equations and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 193))

Abstract

The paper is devoted to studying Banach algebras of shift-invariant singular integral operators with slowly oscillating coefficients and their extensions by shift operators associated with iterations of a slowly oscillating Carleman shift generating a finite cyclic group. Both algebras are contained in the Banach algebra of bounded linear operators on a weighted Lebesgue space with a slowly oscillating Muckenhoupt weight over a composed slowly oscillating Carleson curve. By applying the theory of Mellin pseudodifferential operators, Fredholm symbol calculi for these algebras and Fredholm criteria and index formulas for their elements are established in terms of their Fredholm symbols.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Böttcher and Yu.I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Progress in Mathematics 154, Birkhäuser, Basel, 1997.

    MATH  Google Scholar 

  2. A. Böttcher, Yu.I. Karlovich, and V.S. Rabinovich, Mellin pseudodifferential operators with slowly varying symbols and singular integral on Carleson curves with Muckenhoupt weights. Manuscripta Math. 95 (1998), 363–376.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Böttcher, Yu.I. Karlovich, and V.S. Rabinovich, The method of limit operators for one-dimensional singular integrals with slowly oscillating data. J. Operator Theory 43 (2000), 171–198.

    MATH  MathSciNet  Google Scholar 

  4. A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, 2nd edition. Springer, Berlin, 2006.

    MATH  Google Scholar 

  5. G. David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe. Ann. Sci. École Norm. Sup. 17 (1984), 157–189.

    MATH  Google Scholar 

  6. E.M. Dynkin and B.P. Osilenker, Weighted norm estimates for singular integrals and their applications. J. Soviet Math. 30 (1985), 2094–2154.

    Article  Google Scholar 

  7. N.B. Haaser and J.A. Sullivan, Real Analysis. Dover Publications, New York, 1991.

    Google Scholar 

  8. R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Amer. Math. Soc. 176 (1973), 227–251.

    Article  MATH  MathSciNet  Google Scholar 

  9. N. Karapetiants and S. Samko, Equations with Involutive Operators. Birkhäuser, Boston, 2001.

    MATH  Google Scholar 

  10. A.Yu. Karlovich, Yu.I. Karlovich, and A.B. Lebre, Invertibility of functional operators with slowly oscillating non-Carleman shifts. Operator Theory: Advances and Applications 142 (2003), 147–174.

    MathSciNet  Google Scholar 

  11. Yu.I. Karlovich, An algebra of pseudodifferential operators with slowly oscillating symbols. Proc. London Math. Soc. 92 (2006), 713–761.

    Article  MATH  MathSciNet  Google Scholar 

  12. Yu.I. Karlovich, Pseudodifferential operators with compound slowly oscillating symbols. Operator Theory: Advances and Applications 171 (2007), 189–224.

    Article  MathSciNet  Google Scholar 

  13. Yu.I. Karlovich, Algebras of pseudo-differential operators with discontinuous symbols. Operator Theory: Advances and Applications 172 (2007), 207–233.

    Article  MathSciNet  Google Scholar 

  14. Yu.I. Karlovich, Nonlocal singular integral operators with slowly oscillating symbols. Operator Theory: Advances and Applications 181 (2008), 229–261.

    Article  Google Scholar 

  15. Yu.I. Karlovich, The Haseman boundary value problem with slowly oscillating data. Analytic Methods of Analysis and Differential Equations: AMADE 2006, Cambridge, Cambridge Scientific Publishers, 2008, pp. 81–110.

    Google Scholar 

  16. Yu.I. Karlovich and E. Ramírez de Arellano, Singular integral operators with fixed singularities on weighted Lebesgue spaces. Integral Equations and Operator Theory 48 (2004), 331–363.

    Article  MATH  MathSciNet  Google Scholar 

  17. G.S. Litvinchuk, Boundary Value Problems and Singular Integral Equations with Shift. Nauka, Moscow, 1977 [Russian].

    MATH  Google Scholar 

  18. V.S. Rabinovich, Mellin pseudodifferential operators techniques in the theory of singular integral operators on some Carleson curves. Operator Theory: Advances and Applications 102 (1998), 201–218.

    MathSciNet  Google Scholar 

  19. V. Rabinovich, S. Roch, and B. Silbermann, Limit Operators and Their Applications in Operator Theory. Birkhäuser, Basel, 2004.

    MATH  Google Scholar 

  20. D. Sarason, Toeplitz operators with piecewise quasicontinuous symbols. Indiana Univ. Math. J. 26 (1977), 817–838.

    Article  MATH  MathSciNet  Google Scholar 

  21. I.B. Simonenko and Chin Ngok Min, Local Method in the Theory of One-dimensional Singular Integral Equations with Piecewise Continuous Coefficients. Noetherity. University Press, Rostov on Don, 1986 (Russian).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Col. Chamilpa, C.P. 62209, Cuernavaca, Morelos, México

    Yu. I. Karlovich

Authors
  1. Yu. I. Karlovich
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università della Basilicata, Viale dell’Ateneo Lucano 10, 85100, Potenza, Italy

    Alberto Cialdea

  2. Dipartimento di Matematico “Guido Castelnuovo”, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185, Rome, Italy

    Paolo Emilio Ricci  & Flavia Lanzara  & 

Additional information

To Professor V.G. Maz’ya on the occasion of his 70th birthday

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Karlovich, Y.I. (2009). An Algebra of Shift-invariant Singular Integral Operators with Slowly Oscillating Data and Its Application to Operators with a Carleman Shift. In: Cialdea, A., Ricci, P.E., Lanzara, F. (eds) Analysis, Partial Differential Equations and Applications. Operator Theory: Advances and Applications, vol 193. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9898-9_8

Download citation

Publish with us

Policies and ethics

Search

Navigation