Directional quaternionic hilbert operators

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 (equation presented) Let J1, J2 be the complex structures on the tangent bundle of H ∼ C2 defined by left multiplication by i and j. Let J∗1, J∗2 be the dual structures on the cotangent bundle and set J∗3 = J∗1 J∗2 . For every complex structure Jp = p1J1+p2J2+p3J3 (p ε S2 an imaginary unit), let ∂p = 1/2 (d + pJ∗p° d) be the Cauchy-Riemann operator w.r.t. the structure Jp. Let Cp = 1, p ∼ C. If Δ satisfies a geometric condition, for every Cp-valued function f1 in a Sobolev space on the boundary ∂Δ, we obtain a function Hp(f1) : ∂Δ → Cp , such that f = f1 + Hp(f1) is the trace of a regular function on Δ. The function Hp(f1) is uniquely characterized by L2 (∂Δ)-orthogonality to the space of CR functions w.r.t. the structure Jp. In this way we get, for every direction p ε S2, a bounded linear Hilbert operator Hp, with the property that H2p = id - Sp, where Sp is the Szegö projection w.r.t. the structure Jp.