On Characterization of Hilbert Transform of Riemannian Surface with Boundary

Let (M, g) be a smooth compact orientable two-dimensional Riemannian manifold (surface) with a smooth metric tensor g and a smooth connected boundary Gamma. The Hilbert transform H associated with (M, g) acts in C(Gamma; R) by H : R(eta )bar right arrow S-N, where eta = omega vertical bar Gamma is the trace of a function omega holomorphic in M. We provide characteristic conditions on an operator H defined on a curve Gamma to be the Hilbert transform of a surface. In fact, the characterization of H is reduced to one of the Dirichlet-to-Neumann map Lambda(g) of the surface (M, g), which is related to the Hilbert transform by H = J Lambda(g), where J is integration along Gamma. In contrast to the known characterization of Lambda(g) by Henkin and Michel in terms of multidimensional complex analysis, our one makes use of the Commutative Banach Algebra theory.