An Atomic Representation for Bicomplex Hardy Classes
William L. Blair
TL;DR
The paper develops a bicomplex analogue of holomorphic Hardy spaces on the unit disk by relating bicomplex-valued Hardy functions to complex Hardy spaces through an idempotent decomposition, enabling a representation of $f \in H^p(D,\mathbb{B})$ as $f = p^+ f^+ + p^- f^-$ with $(f^+)^*, f^- \in H^p(D)$. It shows that bicomplex Hardy functions have boundary values in the sense of distributions, and these boundary values admit an atomic decomposition arising from the complex theory; the Hilbert transform is extended to these boundary distributions. The work then generalizes to bicomplex generalized Hardy spaces $H^p_w(D,\mathbb{B})$ with a representation $f = \varphi + T_{\mathbb{B}}(w)$ and corresponding boundary behavior, and finally extends the framework to higher-order bicomplex Hardy spaces $H^{n,p}_{w}(D,\mathbb{B})$ via nested Cauchy-type operators and kernels, together with their boundary properties and Hilbert-transform stability. Collectively, these results bridge bicomplex analysis with classical Hardy space theory, yielding robust boundary-distribution representations and operator-continuity results that persist under generalizations and higher-order equations.
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.