# Directional quaternionic hilbert operators

• Alessandro Perotti
Conference paper
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

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