Hilbert transform for the three-dimensional Vekua equation

Abstract

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.