An Atomic Representation for Bicomplex Hardy Classes

TL;DR

This paper introduces a new atomic representation for bicomplex Hardy class functions, extending complex Hardy space theory to bicomplex analysis, and explores their boundary behavior and Hilbert transform properties.

Contribution

It develops an atomic decomposition framework for bicomplex Hardy class functions and analyzes their boundary values and Hilbert transform continuity.

Findings

Bicomplex Hardy functions have boundary values as distributions.

These boundary values admit an atomic decomposition.

The Hilbert transform is continuous on these distributional boundary values.

Abstract

We develop representations for bicomplex-valued functions in Hardy classes that generalize the complex holomorphic Hardy spaces. Using these representations, we show these functions have boundary values in the sense of distributions that are representable by an atomic decomposition, and we show continuity of the Hilbert transform on this class of distributional boundary values.