Skip to main content
SpringerLink
Log in
Book cover

Analysis of Divergence pp 119–143Cite as

Convolution Calderón-Zygmund singular integral operators with rough kernels

  • Chapter

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

A survey of known results in the theory of convolution type Calderón-Zygmund singular integral operators with rough kernels is given. Some recent progress is discussed. A list of remaining open questions is presented.

Research partially supported by NSF grant DMS 9623120 and by the University of Missouri Research Board

This is a preview of subscription content, 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   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952) p. 85–139.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956) p.289–309.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Chang, S. Krantz, and E. M. Stein, Hardy spaces and elliptic boundary value problems, Cont. Math. 137 (1992) p.119–131.

    MathSciNet  Google Scholar 

  4. M. Christ, Weak type (1,1) bounds for rough operators, Ann. Math. 128 (1988) p. 19–42.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Christ and J.-L. Rubio de Francia, Weak type (1,1) bounds for rough operators II, Invent. Math. 93 (1988) P.225–237.

    Article  MathSciNet  MATH  Google Scholar 

  6. R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977) p.569–645.

    Article  MathSciNet  MATH  Google Scholar 

  7. W. C. Connett, Singular integrals near L 1, Proc. Sympos. Pure Math., Amer. Math. Soc. 35 (1979) p. 163–165 (S. Wainger and G. Weiss, eds.).

    Google Scholar 

  8. J. E. Daly and K. Phillips, On the classification of homogeneous multipliers bounded on H 1(R 2), Proc. Amer. Math. Soc. 106 (1989) p.685–696.

    MathSciNet  MATH  Google Scholar 

  9. J. Duonandikoetxea and J.-L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986) p.541–561.

    Article  MathSciNet  Google Scholar 

  10. D. Fan and Y. Pan, Singular integral operators with rough kernels supported by subvarieties, Amer. J. Math. 119 (1997) p.799–839.

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Fefferman, A theory of entropy in Fourier analysis, Adv. Math. 30 (1978) p.171–201.

    Article  MathSciNet  MATH  Google Scholar 

  12. L. Grafakos and A. Stefanov, L p bounds for singular integrals and maximal singular integrals with rough kernels, (1997), to appear, Indiana Univ. Math. J. (1998).

    Google Scholar 

  13. S. Hofmann, Weak (1,1) boundedness of singular integrals with nonsmooth kernels, Proc. Amer. Math. Soc. 103 (1988) p.260–264.

    MathSciNet  MATH  Google Scholar 

  14. F. Ricci and G. Weiss, A characterization of H l(∑ n_1, Proc. Sympos. Pure Math, Amer. Math. Soc. 35 (1979) p.289–294.

    MathSciNet  Google Scholar 

  15. A. Seeger, Singular integral operators with rough convolution kernels, J. Amer. Math. Soc. 9 (1996) p.95–105.

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Seeger and T. Tao, personal communication.

    Google Scholar 

  17. P. Sjögren and F. Soria, Rough maximal operators and rough singular integral operators applied to integral radial functions, Rev. Math. Iber. 13 (1997) p1–18.

    MATH  Google Scholar 

  18. A. Stefanov, Characterizations of H 1 and applications to singular integral operators applied to integral radial functions, submitted.

    Google Scholar 

  19. E. M. Stein, Singular Integrals and Diffemetiability Properties of Functions, Princeton University Press (1970).

    Google Scholar 

  20. E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press (1993).

    MATH  Google Scholar 

  21. M. Weiss and A. Zygmund, An example in the theory of singular integrals, Studia Math. 26 (1965) p.101–111.

    MathSciNet  MATH  Google Scholar 

  22. A. Zygmund, Trigonometric Series, Vol I, Cambridge University Press (1959).

    MATH  Google Scholar 

Download references

Authors
  1. L. Grafakos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Stefanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, University of Maine, Orono, ME, 04469, USA

    William O. Bray

  2. Department of Mathematics and Statistics, University of Missouri, Rolla Bldg 202, Rolla, MO, 65409, USA

    Časlav V. Stanojević

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Birkhäuser Boston

About this chapter

Cite this chapter

Grafakos, L., Stefanov, A. (1999). Convolution Calderón-Zygmund singular integral operators with rough kernels. In: Bray, W.O., Stanojević, Č.V. (eds) Analysis of Divergence. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2236-1_10

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-2236-1_10

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-7467-4

  • Online ISBN: 978-1-4612-2236-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation