Hilbert transform for the three-dimensional Vekua equation

TL;DR

This paper extends the concept of the Hilbert transform to three dimensions within the framework of the Vekua equation, linking boundary data to quaternionic monogenic functions and exploring related boundary value problems.

Contribution

It generalizes the 3D Hilbert transform for the Vekua equation, connecting boundary data with quaternionic monogenic functions and analyzing the Dirichlet-to-Neumann map in R3.

Findings

Defined the 3D Hilbert transform for the Vekua equation

Established boundary correspondence for quaternionic monogenic functions

Explored the 3D Dirichlet-to-Neumann map for conductivity

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.