Maximal Function Characterizations of Hardy Spaces on ${\mathbb{R}}^{n}$ with Pointwise Variable Anisotropy

TL;DR

This paper extends the characterization of Hardy spaces with pointwise variable anisotropy on ${ m f R}^n$ using maximal functions, generalizing previous anisotropic Hardy space results.

Contribution

It provides new real-variable characterizations of Hardy spaces $H^p( heta)$ with pointwise continuous covers, broadening the understanding of anisotropic Hardy spaces.

Findings

Characterizations via radial, non-tangential, and tangential maximal functions.

Generalization of known results on anisotropic Hardy spaces.

Applicable to covers with high anisotropy and pointwise continuity.

Abstract

In 2011, Dekel et al. developed highly geometric Hardy spaces $H^p(\Theta)$, for the full range $0<p\leq 1$, which are constructed by continuous multi-level ellipsoid covers $\Theta$ of $\mathbb{R}^n$ with high anisotropy in the sense that the ellipsoids can change shape rapidly from point to point and from level to level. In this article, if the cover $\Theta$ is pointwise continuous, then the authors further obtain some real-variable characterizations of $H^p(\Theta)$ in terms of the radial, the non-tangential and the tangential maximal functions, which generalize the known results on the anisotropic Hardy spaces of Bownik.