Characterizations of Hardy spaces on tube domains over polyhedral cones

TL;DR

This paper characterizes Hardy spaces on tube domains over polyhedral cones, introducing new integral formulas and multi-parameter analysis tools to handle boundary values and geometric complexities.

Contribution

It develops a novel iterated Poisson integral formula and adapts multi-parameter harmonic analysis techniques for Hardy spaces on polyhedral cone tube domains.

Findings

Established equivalence of various Hardy space characterizations.

Introduced new geometric notions like twisted rectangles and non-tangential regions.

Connected boundary values with non-tangential maximal and Littlewood-Paley functions.

Abstract

This paper is devoted to the equivalence of various characterizations of holomorphic $H^1$ Hardy spaces on tube domains over polyhedral cones. We establish a new iterated Poisson integral formula which reproduces holomorphic functions on such domains. However, this formula shows that holomorphic $H^1$ functions have boundary values in a new type of Hardy space of real variables on their Shilov boundaries $\mathbb{R}^n$, which cannot be treated by standard classical multi-parameter harmonic analysis. We overcome this difficulty by developing techniques suitably adapted in this setting. Using the iterated Poisson integral as our approximation to the identity, and employing a lifting technique, we introduce various notions of multi-parameter analysis adapted to tube domains, such as twisted rectangles, new non-tangential approach regions, non-tangential maximal functions and Littlewood-Paley type functions. All these notions exhibit new geometric features associated with polyhedral cones and involve hidden parameters, as in the flag setting. We develop the necessary multi-parameter tools to investigate these new Hardy spaces. In particular, we apply these tools to obtain equivalent characterizations of the holomorphic $H^1$ Hardy spaces on tube domains in terms of non-tangential maximal, Lusin-Littlewood-Paley area and Littlewood-Paley $g$-functions.