Skip to main content
SpringerLink
Log in

Approximate dimension applied to criteria for monogenicity on fractal domains

  • Published:
Bulletin of the Brazilian Mathematical Society, New Series Aims and scope Submit manuscript

Abstract

Suppose that Ω is a bounded domain of ℝn with a fractal boundary Γ and let ℝ0,n be the real Clifford algebra constructed over the quadratic space ℝn. Replacing the fractal dimensions of Γ with conditions of approximating character we will characterize the monogenicity of a ℝ0,n -valued function F in the interior and exterior of Ω, in terms of its boundary value f = F|Γ. Moreover, our geometric perspective allows for generalizations of certain two-sided monogenic extension results to a wide class of domains.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Abreu Blaya, J. Bory Reyes and B. Kats. On the solvability of the jump problem in Clifford analysis. submitted.

  2. R. Abreu Blaya, J. Bory Reyes and T. Moreno García. Cauchy Transform on nonrectifiable surfaces in Clifford Analysis. J. Math. Anal. Appl., 339 (2008), 31–44.

    Article  MATH  MathSciNet  Google Scholar 

  3. R. Abreu Blaya, J. Bory Reyes and T. Moreno García. Teodorescu transform decomposition of multivector fields on fractal hypersurfaces Oper. TheoryAdv. Appl., 167 (2006), 1–16.

    Article  Google Scholar 

  4. R. Abreu Blaya, J. Bory Reyes and D. Peña Peña. Jump problem and removable singularities for monogenic functions. J. Geom. Anal., 17(1) (2007), March, 1–14.

    Article  MATH  MathSciNet  Google Scholar 

  5. R. Abreu Blaya and J. Bory Reyes. Criteria for monogenicity of Clifford algebravalued functions on fractal domain. Arch. Math. (Basel), 95(1) (2010), 45–51.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Abreu Blaya and J. Bory Reyes. Hölder norm estimate for the Hilbert transform in Clifford analysis. Bull. Braz. Math. Soc., 41(3) (2010), 389–398.

    Article  MATH  MathSciNet  Google Scholar 

  7. F. Brackx, R. Delanghe and F. Sommen. Clifford analysis. Research Notes in Mathematics, 76, Pitman, Boston (1982).

    MATH  Google Scholar 

  8. H. Federer. Geometric measure theory. Die Grundlehren der mathematischenWissenschaften, Band 153, Springer-Verlag New York Inc., New York (1969).

    MATH  Google Scholar 

  9. K. Gürlebeck, K. Habetha and W. Sprössig. Holomorphic functions in the plane and n-dimensional space, translated from the 2006 German original. Birkh.user Verlag, Basel (2008).

    MATH  Google Scholar 

  10. J. Harrison and A. Norton. The Gauss-Green theorem for fractal boundaries. Duke Mathematical Journal, 67(3) (1992), 575–588.

    Article  MATH  MathSciNet  Google Scholar 

  11. B. Kats. On solvability of the jump problem. J. Math. Anal. Appl., 356(2) (2009), 577–581.

    Article  MATH  MathSciNet  Google Scholar 

  12. M.L. Lapidus and H. Maier. Hypothése de Riemann, cordes fractales vibrantes et conjecture de Weyl-Berry modifiée. C.R. Acad. Sci. Paris Série I Math., 313(1) (1991), 19–24.

    MATH  MathSciNet  Google Scholar 

  13. E.M. Stein. Singular Integrals and Differentiability Properties of Functions. Princenton Math. Ser., 30, Princenton Univ. Press, Princenton, N.J. (1970).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Facultad de Informática y Matemática, Universidad de Holguín, Holguín, 80100, Cuba

    Ricardo Abreu-Blaya

  2. Departamento de Matemática, Universidad de Oriente, Santiago de Cuba, 90500, Cuba

    Juan Bory-Reyes

  3. Chair of Mathematics Kazan State University of Architecture and Engineering, Zelenaya Street, 1, Kazan Tatarstan, 420043, Russia

    Boris A. Kats

Authors
  1. Ricardo Abreu-Blaya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juan Bory-Reyes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Boris A. Kats
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ricardo Abreu-Blaya.

About this article

Cite this article

Abreu-Blaya, R., Bory-Reyes, J. & Kats, B.A. Approximate dimension applied to criteria for monogenicity on fractal domains. Bull Braz Math Soc, New Series 43, 529–544 (2012). https://doi.org/10.1007/s00574-012-0025-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00574-012-0025-z

Keywords

Mathematical subject classification

Search

Navigation