Szeg$\ddot{o}$ projection and matrix Hilbert transform in Hermitean Clifford analysis

TL;DR

This paper develops the Szegő projection and matrix Hilbert transform in Hermitean Clifford analysis, establishing their properties, connections, and characterizations within the Hardy space framework for Hermitean monogenic functions.

Contribution

It introduces the Szegő projection operator for Hermitean monogenic functions, establishes the Kerzman-Stein formula, and characterizes the unitarity of the matrix Hilbert transform in this setting.

Findings

Szegő projection operator expressed via Hardy projection and its adjoint.

Kerzman-Stein formula connecting Szegő and Hardy projections.

Algebraic and geometric criteria for unitarity of the matrix Hilbert transform.

Abstract

The simultaneous null solutions of the two complex Hermitean Dirac operators are focused on in Hermitean Clifford analysis, where the matrix Hilbert transform was presented and proved to satisfy the analogous properties of the Hilbert transform in classical analysis and in orthogonal Clifford analysis. Under this setting we will introduce the Szeg$\ddot{o}$ projection operator for the Hardy space of Hermitean monogenic functions defined on a bounded subdomain of even dimensional Euclidean space, establish the Kerzman-Stein formula which closely connects the Szeg$\ddot{o}$ projection operator with the Hardy projection operator onto the Hardy space of Hermitean monogenic functions defined on a bounded subdomain of even dimensional Euclidean space, and get the Szeg$\ddot{o}$ projection operator in terms of the Hardy projection operator and its adjoint. Further we will give the algebraic and geometric characterizations for the matrix Hilbert transform to be unitary in Hermitean Clifford analysis.