  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.
