Directional quaternionic hilbert operators

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


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.


Quaternionic regular function hyperholomorphic function Hilbert operator conjugate harmonic 

Mathematics Subject Classification (2000)

Primary 30G35 Secondary 32A30 


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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

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

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