Advertisement

Directional quaternionic hilbert operators

  • Alessandro Perotti
Part of the Trends in Mathematics book series (TM)

Abstract

The paper discusses harmonic conjugate functions and Hilbert operators in the space of Fueter regular functions of one quaternionic variable. We consider left-regular functions in the kernel of the Cauchy-Riemann operator
$$ D = 2\left( {\frac{\partial } {{\partial \bar z_1 }} + j\frac{\partial } {{\partial \bar z_2 }}} \right) = \frac{\partial } {{\partial x_0 }} + i\frac{\partial } {{\partial x_1 }} + j\frac{\partial } {{\partial x_2 }} - k\frac{\partial } {{\partial x_3 }}. $$
Let J1, J2 be the complex structures on the tangent bundle of ” ≃ ℂ2 defined by left multiplication by i and j. Let J1 *, J2 * be the dual structures on the cotangent bundle and set J3 * = J1 *J2 *. For every complex structure J p = p 1 J 1 + p 2 J 2 +p 3 J 3 (pSan imaginary unit), let \( \bar \partial p = \tfrac{1} {2}(d + pJ_p^* \circ d) \) be the Cauchy-Riemann operator w.r.t. the structure Jp. Let ℂ p = 〈1,p〉 ≃ ℂ. If Ω satisfies a geometric condition, for every ℂ-valued function f1 in a Sobolev space on the boundary ∂Ώ, we obtain a function Hp(f1): ∂Ώ → ℂp, such that f = f 1 + H p (f 1) is the trace of a regular function on Ώ. The function H p (f 1) is uniquely characterized by L2(∂Ώ)-orthogonality to the space of CR-functions w.r.t. the structure J p . In this way we get, for every direction pS 2, a bounded linear Hilbert operator Hp, with the property that H p = id − S p , where S p is the Szegö projection w.r.t. the structure J p.

Keywords

Quaternionic regular function hyperholomorphic function Hilbert operator conjugate harmonic 

Mathematics Subject Classification (2000)

Primary 30G35 Secondary 32A30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. Abreu-Blaya, J. Bory-Reyes, M. Shapiro, On the notion of the Bochner-Martinelli integral for domains with rectifiable boundary. Complex Anal. Oper. Theory 1 (2007), no. 2, 143–168.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    H. Begehr, Complex analytic methods for partial differential equations, ZAMM 76 (1996), Suppl. 2, 21–24.zbMATHGoogle Scholar
  3. [3]
    H. Begehr, Boundary value problems in ℂ and ℂn, Acta Math. Vietnam. 22 (1997), 407–425.zbMATHMathSciNetGoogle Scholar
  4. [4]
    H. Begehr and A. Dzhuraev, An Introduction to Several Complex Variables and Partial Differential Equations, Addison Wesley Longman, Harlow, 1997.zbMATHGoogle Scholar
  5. [5]
    H. Begehr and A. Dzhuraev, Overdetermined systems of second order elliptic equations in several complex variables. In: Generalized analytic functions (Graz, 1997), Int. Soc. Anal. Appl. Comput., 1, Kluwer Acad. Publ., Dordrecht, 1998, pp. 89–109.Google Scholar
  6. [6]
    S. Bell, The Cauchy transform, potential theory and conformal mapping. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.Google Scholar
  7. [7]
    F. Brackx, R. Delanghe, F. Sommen, On conjugate harmonic functions in Euclidean space. Clifford analysis in applications. Math. Methods Appl. Sci. 25 (2002), no. 16–18, 1553–1562.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    F. Brackx, B. De Knock, H. De Schepper and D. Eelbode, On the interplay between the Hilbert transform and conjugate harmonic functions. Math. Methods Appl. Sci. 29 (2006), no. 12, 1435–1450.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    F. Brackx, H. De Schepper and D. Eelbode, A new Hilbert transform on the unit sphere in ℝm. Complex Var. Elliptic Equ. 51 (2006), no. 5–6, 453–462.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    J. Chen and J. Li, Quaternionic maps between hyperkhler manifolds, J. Differential Geom. 55 (2000), 355–384.zbMATHMathSciNetGoogle Scholar
  11. [11]
    A. Cialdea, On the Dirichlet and Neumann problems for pluriharmonic functions. In: Homage to Gaetano Fichera, Quad. Mat., 7, Dept. Math., Seconda Univ. Napoli, Caserta, 2000, pp. 31–78.Google Scholar
  12. [12]
    R. Delanghe, On some properties of the Hilbert transform in Euclidean space. Bull. Belg. Math. Soc. Simon Stevin 11 (2004), no. 2, 163–180.zbMATHMathSciNetGoogle Scholar
  13. [13]
    J. Detraz, Problème de Dirichlet pour le système ∂2 f/∂zi∂zj = 0. (French), Ark. Mat. 26 (1988), no. 2, 173–184.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    A. Dzhuraev, On linear boundary value problems in the unit ball of ℂn, J. Math. Sci. Univ. Tokyo 3 (1996), 271–295.zbMATHMathSciNetGoogle Scholar
  15. [15]
    A. Dzhuraev, Some boundary value problems for second order overdetermined elliptic systems in the unit ball of ℂn. In: Partial Differential and Integral Equations (eds.: H. Begehr et al.), Int. Soc. Anal. Appl. Comput., 2, Kluwer Acad. Publ., Dordrecht, 1999, pp. 37–57.Google Scholar
  16. [16]
    G. Fichera, Alcuni recenti sviluppi della teoria dei problemi al contorno per le equazioni alle derivate parziali lineari. (Italian) In: Convegno Internazionale sulle Equazioni Lineari alle Derivate Parziali, Trieste, 1954, Edizioni Cremonese, Roma, 1955, pp. 174–227.Google Scholar
  17. [17]
    G. Fichera, Linear elliptic differential systems and eigenvalue problems. Lecture Notes in Mathematics, 8 Springer-Verlag, Berlin-New York, 1965.Google Scholar
  18. [18]
    K. Gürlebeck, K. Habetha and W. Sprössig, Holomorphic Functions in the Plane and n-dimensional Space. Birkhäuser, Basel, 2008.zbMATHGoogle Scholar
  19. [19]
    K. Gürlebeck and W. Sprössig, Quaternionic Analysis and Elliptic Boundary Value Problems. Birkhäuser, Basel, 1990.zbMATHGoogle Scholar
  20. [20]
    D. Joyce, Hypercomplex algebraic geometry, Quart. J. Math. Oxford 49 (1998), 129–162.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    V.V. Kravchenko and M.V. Shapiro, Integral representations for spatial models of mathematical physics, Harlow: Longman, 1996.zbMATHGoogle Scholar
  22. [22]
    A.M. Kytmanov, The Bochner-Martinelli integral and its applications, Birkhäuser Verlag, Basel, 1995.zbMATHGoogle Scholar
  23. [23]
    M. Naser, Hyperholomorphe Funktionen, Sib. Mat. Zh. 12, 1327–1340 (Russian). English transl. in Sib. Math. J. 12, (1971) 959-968.zbMATHGoogle Scholar
  24. [24]
    K. Nono, α-hyperholomorphic function theory, Bull. Fukuoka Univ. Ed. III 35 (1985), 11–17.zbMATHMathSciNetGoogle Scholar
  25. [25]
    K. Nōno, Characterization of domains of holomorphy by the existence of hyperharmonic functions, Rev. Roumaine Math. Pures Appl. 31 n. 2 (1986), 159–161.zbMATHMathSciNetGoogle Scholar
  26. [26]
    A. Perotti, Dirichlet Problem for pluriharmonic functions of several complex variables, Communications in Partial Differential Equations, 24, nn. 3&4, (1999), 707–717.zbMATHMathSciNetGoogle Scholar
  27. [27]
    A. Perotti, Holomorphic functions and regular quaternionic functions on the hyperkähler space ℍ. In: More Progresses in Analysis: Proceedings of the 5th international ISAAC Congress, (Catania 2005), eds. H.G.W. Begehr, F. Nicolosi, World Scientific, Singapore, 2008.Google Scholar
  28. [28]
    A. Perotti, Quaternionic regular functions and the ∂-Neumann problem in ℂ2, Complex Variables and Elliptic Equations 52 No. 5 (2007), 439–453.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    A. Perotti, Dirichlet problem for pluriholomorphic functions of two complex vari-ables, J. Math. Anal. Appl. 337/1 (2008), 107–115.CrossRefMathSciNetGoogle Scholar
  30. [30]
    A. Perotti, Every biregular function is biholomorphic, Advances in Applied Clifford Algebras, in press.Google Scholar
  31. [31]
    T. Qian, Analytic signals and harmonic measures. J. Math. Anal. Appl. 314 (2006), no. 2, 526–536.CrossRefMathSciNetGoogle Scholar
  32. [32]
    R. Rocha-Chavez, M.V. Shapiro, L.M. Tovar Sanchez, On the Hilbert operator for α-hyperholomorphic function theory in ℝ2. Complex Var. Theory Appl. 43, no. 1 (2000), 1–28.zbMATHMathSciNetGoogle Scholar
  33. [33]
    W. Rudin, Function theory in the unit ball ofn, Springer-Verlag, New York, Heidelberg, Berlin 1980.Google Scholar
  34. [34]
    M.V. Shapiro and N.L. Vasilevski, Quaternionic ψ-hyperholomorphic functions, singular integral operators and boundary value problems. I. ψ-hyperholomorphic function theory, Complex Variables Theory Appl. 27 no.1 (1995), 17–46.zbMATHMathSciNetGoogle Scholar
  35. [35]
    M.V. Shapiro and N.L. Vasilevski, Quaternionic Ψ-hyperholomorphic functions, singular integral operators and boundary value problems. II: Algebras of singular integral operators and Riemann type boundary value problems, Complex Variables Theory Appl. 27 no.1 (1995), 67–96.zbMATHMathSciNetGoogle Scholar
  36. [36]
    E.L. Stout, Hp-functions on strictly pseudoconvex domains, Amer. J. Math. 98 n.3 (1976), 821–852.zbMATHCrossRefMathSciNetGoogle Scholar
  37. [37]
    A. Sudbery, Quaternionic analysis, Mat. Proc. Camb. Phil. Soc. 85 (1979), 199–225.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Alessandro Perotti
    • 1
  1. 1.Department of MathematicsUniversity of TrentoPovo TrentoItaly

Personalised recommendations