  The three-dimensional Hilbert transform takes scalar data on the boundary of
a domain in R3 and produces the boundary value of the vector part of a
quaternionic monogenic (hyperholomorphic) function of three real variables, for
which the scalar part coincides with the original data. This is analogous to
the question of the boundary correspondence of harmonic conjugates.
Generalizing a representation of the Hilbert transform H in R3 given by T. Qian
and Y. Yang (valid in Rn), we define the Hilbert transform Hf associated to the
main Vekua equation DW = (Df/f)W in bounded Lipschitz domains in R3. This leads
to an investigation of the three-dimensional analogue of the
Dirichlet-to-Neumann map for the conductivity equation.
