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 Cauchy-Riemann operator D = 2(partial derivative/partial derivative(z) over bar (1) + j partial derivative/partial derivative(z) over bar (2)) = partial derivative/partial derivative x(0) + i partial derivative/partial derivative x(1) + j partial derivative/partial derivative x(2) - k partial derivative/partial derivative x(3). Let J(1), J(2) be the complex structures on the tangent bundle of H similar or equal to C(2) 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 J(p) = p(1)J(1)+p(2)J(2)+p(3)J(3) (p is an element of S(2) an imaginary unit), let partial derivative(p) = 1/2 (d + pJ(p)* o d) be the Cauchy-Riemann operator w.r.t. tire structure J(p). Let C(p) = < 1, p > similar or equal to C. If Omega satisfies a geometric condition, for every C(p)-valued function f(1) in a Sobolev space on the boundary partial derivative Omega, we obtain a function H(p)(f(1)) : partial derivative Omega --> C(p)(perpendicular to), such that f = f(1) + H(p)(f(1)) is the trace of a regular function on Omega. The function H(p)(f(1)) is uniquely characterized by L(2)(partial derivative Omega)-orthogonality to the space of CR-functions w.r.t. the structure J(p). In this way we get, for every direction p is an element of S(2), a bounded linear Hilbert operator H(p), with the property that, H(p)(2) = id - S(p), where S(p) is the Szego projection w.r.t. the structure J(p).