Directional quaternionic Hilbert operators

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 modified Cauchy--Riemann operator D. Let J1,J2 be the complex structures on the tangent bundle of H=C^2 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=p1 J1 + p2 J2 + p3 J3 (p in S^2 an imaginary unit), let dibar_p=1/2 (d + p Jp* o d) be the Cauchy--Riemann operator w.r.t. the structure Jp. Let Cp be the subalgebra generated by p. If O satisfies a geometric condition, for every Cp-valued function f1 in a Sobolev space on the boundary bO, we obtain a function Hp(f1) : bO \rightarrow Cp^{\bot}, such that f=f1+Hp(f1) is the trace of a regular function on O. The function Hp(f1) is uniquely characterized by L^2(bO)-orthogonality to the space of CR-functions w.r.t. the structure Jp. In this way we get, for every direction p in S^2, a bounded linear Hilbert operator Hp, with the property that Hp^2 = id - Sp, where Sp is the Szego projection w.r.t. the structure Jp.