LAUSR

ABSTRACT The three-dimensional Hilbert transform takes scalar data on the boundary of a domain 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 in given by T. Qian and Y. Yang (valid in ), we define the Hilbert transform associated to the main Vekua equation in bounded Lipschitz domains in . This leads to an investigation of the three-dimensional analogue of the Dirichlet-to-Neumann map for the conductivity equation.