Let $(X,\mathbf{q},\mu)$ be an ultra-RD-space
with upper dimension $n\in(0,\infty)$;
i.e., it is a quasi-ultrametric space of homogeneous type whose
measure $\mu$ satisfies an additional reverse doubling property.
Let $\mathrm{ind\,}(X,\mathbf{q})\in(0,\infty]$ denote
its lower smoothness index, as
introduced by Mitrea et al.
In this monograph, the authors first construct a new
approximation of the identity
on quasi-ultrametric
spaces of homogeneous type,
achieving a maximal degree of smoothness
$0<\varepsilon\preceq\mathrm{ind\,}(X,\mathbf{q})$.
This fundamental tool is then used to
derive sharp homogeneous (as well as inhomogeneous)
continuous/discrete Calderón reproducing formulae
on ultra-RD-spaces. As applications,
the authors establish Littlewood-Paley function
characterizations for both Hardy spaces and Triebel-Lizorkin
spaces on ultra-RD-spaces.
The authors further introduce
Hardy-Lorentz spaces $H^{p,q}_\ast(X)$ via
the grand maximal function, with the sharp range
$p\in(\frac{n}{n+\mathrm{ind\,}(X,\mathbf{q})},\infty)$
and $q\in(0,\infty]$, and provide their
real-variable characterizations
using radial/non-tangential
maximal functions, (finite) atoms, molecules,
and various Littlewood-Paley functions.
Based on these characterizations, the authors prove
a duality theorem between Hardy-Lorentz spaces and
Campanato-Lorentz spaces, establish a real interpolation
theorem for Hardy-Lorentz spaces, and
derive boundedness results for Calderón-Zygmund operators
on them.
It should be emphasized
that many of the main results in
this monograph are indeed established in the more general setting of
quasi-ultrametric spaces of homogeneous type.
The novelty of this monograph resides in two key aspects.
First, the assumptions on the underlying space are minimal:
the authors only assume that $\mu$ is Borel-semiregular,
that singletons may have strictly positive measure,
and that $\rho$ is a quasi-ultrametric,
which provides a more general theoretic setting
than the Coifman and Weiss notion of spaces of homogeneous type.
Second, the maximal order of smoothness achieved
by the approximation of the identity and
the regularity exponent in the Calderón reproducing
formulae are both sharp,
as they are intrinsically based on the sharp
regularity index $\mathrm{ind\,}(X,\mathbf{q})$
of the quasi-ultrametric.
Given the sharpness of these tools,
the authors are able to develop the real-variable
theory of the Hardy-Lorentz space
$H^{p,q}_\ast(X)$ for optimal
ranges of $p$ and $q$ (which cannot
be obtained in any of the related results
currently available in the literature),
under minimal assumptions on $X$.
All results presented here are new and to appear
in print for the first time.
Designed to be largely self-contained,
this monograph is an invaluable resource for
graduate students and researchers interested
in the frontier of the function space theory
on ultra-RD spaces.