The Szeg\"o metric associated to Hardy spaces of Clifford algebra valued functions and some geometric properties

TL;DR

This paper introduces a Szeg"o metric for Clifford algebra valued monogenic functions, demonstrating its pseudo-invariance under M"obius transformations, negative curvature on bounded domains, and completeness.

Contribution

It extends the Szeg"o metric concept to hypercomplex Clifford algebra functions, revealing geometric properties and invariance under conformal maps.

Findings

Szeg"o metric has pseudo-invariance under M"obius transformations.

The curvature of the Szeg"o metric is always negative on bounded domains.

The Szeg"o metric is complete on bounded domains.

Abstract

In analogy to complex function theory we introduce a Szeg\"o metric in the context of hypercomplex function theory dealing with functions that take values in a Clifford algebra. In particular, we are dealing with Clifford algebra valued functions that are annihilated by the Euclidean Dirac operator in $\mathbb{R}^{m+1}$. These are often called monogenic functions. As a consequence of the isometry between two Hardy spaces of monogenic functions on domains that are related to each other by a conformal map, the generalized Szeg\"o metric turns out to have a pseudo-invariance under M\"obius transformations. This property is crucially applied to show that the curvature of this metric is always negative on bounded domains. Furthermore, it allows us to establish that this metric is complete on bounded domains.