Boundary Spinors and Boundary Values of Holomorphic Functions

Abstract

A classical result in complex analysis which is an important point in the study of the Hilbert transform (and therefore of singular integral transforms) on ℝ, and which is quite relevant to boundary value problems of different kinds is the following:

• Let H be the Hardy space of the upper half plane (i.e. the space of functions holomorphic in the half plane with boundary values in L ₂(ℝ)). If f is the boundary value of a function in H, then F f and ℜf have the same norm in L ₂(ℝ), and the Hilbert transform maps one to the other.

• Here this result is generalized to general (not necessarily simply connected) domains in ℂ. This is done replacing imaginary and real parts of the boundary value with a decomposition in terms of spinor bundles over the boundary. The paper describes these bundles in a straightforward way without reference to abstract bundles. The main idea, using spinor sections on the boundary, originated in the higher-dimensional analogue of complex analysis, Clifford analysis (see [2]).