Real-Variable Theory of Hardy--Lorentz Spaces on Quasi-Ultrametric Spaces of Homogeneous Type with Reverse-Doubling Property

Abstract: Let $(X,\mathbf{q},μ)$ 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 $μ$ 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 $0&lt;\varepsilon\preceq\mathrm{ind\,}(X,\mathbf{q})$. 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 $H<sup>{p,q}_\ast(X)$ 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 paper develops a complete real-variable theory for Hardy–Lorentz spaces by extending classic harmonic analysis tools to ultra-RD spaces.
• It constructs maximally smooth approximations of the identity and sharp Calderón reproducing formulae, enabling precise operator decomposition.
• It provides robust molecular, atomic, and Littlewood–Paley characterizations, achieving optimal parameter ranges and improved endpoint results.

Real-Variable Theory of Hardy–Lorentz Spaces on Quasi-Ultrametric Spaces of Homogeneous Type with Reverse-Doubling Property

Introduction and Context

This monograph, "Real-Variable Theory of Hardy–Lorentz Spaces on Quasi-Ultrametric Spaces of Homogeneous Type with Reverse-Doubling Property" (2604.02822), systematically develops a sharp real-variable function space theory on quasi-ultrametric spaces of homogeneous type—specifically, spaces that satisfy both the classic doubling condition and a reverse-doubling property (ultra-RD-spaces). Building on and sharpening foundational work by Coifman-Weiss, Macías-Segovia, and Mitrea et al., the theory encompasses maximally smooth approximations of the identity, sharp Calderón reproducing formulae, Littlewood-Paley theory, and detailed characterizations of Hardy–Lorentz and Triebel–Lizorkin spaces in this flexible, non-Euclidean setting.

At the core of the analysis is the extension and quantitative refinement of tools that are optimal with respect to the geometric regularity of the space; notably, the smoothness index associated to the quasi-ultrametric structure is employed throughout to establish best possible ranges for the parameters appearing in Hardy and Lorentz scales.

Geometry and Indices of Quasi-Ultrametric Spaces

A central conceptual advance underpinning this work is the precise quantification of the regularity of non-Euclidean (quasi-ultrametric) spaces via the lower smoothness index (or Assouad-type dimension), denoted ${\mathrm{ind\,}(X,\mathbf{q})}$, and the associated analysis of regularized distances. This index allows for an optimal extension of classical tools from $\mathbb{R}^n$ (where, e.g., the Hardy space theory is valid for $p > n/(n+1)$) to much more general metric spaces.

Sharp metrization theorems ensure one can always work with quasi-ultrametrics of maximal regularity, and a detailed catalog of examples elucidates the variability of the smoothness index, demonstrating, for instance, that ultrametric spaces admit real-variable Hardy space theory for the full range $p \in (0,1]$, far surpassing what is possible in standard metric settings.

Maximally Smooth Approximations of the Identity

The authors construct approximations of the identity (AOIs) with the maximal possible Hölder regularity permitted by the geometry (dictated by ${\mathrm{ind\,}(X,\mathbf{q})}$): For any $$0<\varepsilon \preceq {\mathrm{ind\,}}(X,\mathbf{q})$$, there exists a family of symmetric, localized integral operators $\{\mathcal{S}_t\}_{t>0}$ with uniform bounds in norm and Hölder continuity, satisfying homogeneity and support conditions tailored to the quasi-ultrametric structure.

Compared to the literature (e.g., [Alvarado–Mitrea 2015]), these constructions (1) allow positive-measure singletons, (2) function for the full smoothness index, and (3) do not require the regularity of the measure assumed in prior work.

Calderón Reproducing Formulae and Operator Theory

A critical technical tool employed is the establishment of sharp Calderón reproducing formulae—both in homogeneous and inhomogeneous, and continuous and discrete forms—valid on ultra-RD-spaces with Borel-semiregular measures. The authors show that, using their maximally smooth AOIs, one can decompose the identity operator into main and remainder terms with precise control of kernel size and regularity, insuring invertibility on $L^2$ and appropriate spaces of test functions.

This sharply extends results previously restricted to metric, or even Ahlfors-regular, spaces. Intertwined with these results are proofs of boundedness of general Calderón–Zygmund operators on localized test function spaces, with careful tracking of cancellation and support, which are essential for applications to Hardy-type space theory, interpolation, and duality.

Littlewood–Paley Theory and Triebel–Lizorkin Spaces

Leveraging the reproducing formulae, the monograph develops a robust Littlewood–Paley theory in this setting. The authors obtain atomic, molecular, maximal, and (various) $g$-function characterizations for Hardy-type spaces attached to the quasi-ultrametric, as well as sharp Lusin area and $g^*_\lambda$ function characterizations for Triebel–Lizorkin spaces.

Crucially, the endpoints and parameter ranges for these results are dictated by the intrinsic geometry via the smoothness index, leading to maximal admissible ranges, and including critical cases previously inaccessible.

Hardy–Lorentz Spaces: Complete Real-Variable Theory

The culminating achievement is the systematic development of a complete real-variable theory for Hardy–Lorentz spaces $\mathbb{R}^n$0, where $\mathbb{R}^n$1 and $\mathbb{R}^n$2 may be arbitrarily close to $\mathbb{R}^n$3 and $\mathbb{R}^n$4, subject only to minimal geometric restrictions. The theory includes:

• Grand maximal, radial, and non-tangential maximal function definitions and their equivalence,
• Atomic and molecular decompositions for the full optimal range $\mathbb{R}^n$5,
• Littlewood-Paley-based characterizations,
• Real interpolation between $\mathbb{R}^n$6 and $\mathbb{R}^n$7,
• Duality identification with Campanato–Lorentz spaces,
• Boundedness for Calderón–Zygmund operators in (sub)critical cases.

Notable here is that for ultra-RD-spaces with infinite smoothness index (such as ultrametric spaces), one obtains full-range theory for $\mathbb{R}^n$8 and $\mathbb{R}^n$9, which strictly strengthens any prior bound in the general setting. The approach unifies the metric and ultrametric cases, ensuring no artificial restrictions on the exponents.

Strong, quantitative implications are provided: for example, the molecular and Littlewood–Paley characterizations of $p > n/(n+1)$0 on ultra-RD-spaces are universally sharp with respect to smoothness and localization. Contrasts with previous endpoint obstructions (e.g., limitations in [Coifman–Weiss], [Macías–Segovia], [Zhou et al.]) are discussed and resolved.

Implications, Future Directions, and Theoretical Impact

The methods and results of this monograph yield a unified, optimal theory of function spaces and operator theory in geometric measure-theoretic settings far broader than previously accessible, subsuming Euclidean, Carnot–Carathéodory, and fractal (including ultrametric) examples under one analytic umbrella.

Practically, these advances both clarify the structure of endpoint problems and extend the toolkit for handling critical bounds for singular integrals, interpolation, and duality in nonstandard geometries. The applications for analysis and PDEs extend to both classic and modern settings: for instance, fractal and p-adic analysis, or analysis on metric spaces with highly singular measures.

Future developments may include extending theory to weighted/multilinear settings; adapting the atomic and Littlewood–Paley machinery to nonlinear and time-frequency analytic contexts; and deepening the links with geometric group theory, random processes on fractal measure spaces, and emerging topics in data analysis and geometry-aware AI architectures.

Conclusion

This monograph provides a mathematically rigorous, quantitatively sharp, and comprehensive treatment of Hardy–Lorentz and related function spaces on spaces of homogeneous type equipped with quasi-ultrametrics and reverse-doubling measures. By grounding all results in the intrinsic geometric regularity of the space, the authors not only resolve longstanding open questions on admissible exponents and operator boundedness, but also provide a template for future harmonic analysis in abstract and fractal environments. The systematic and modular approach ensures wide applicability and robust generalization capabilities for both theoretical developments and applications in mathematical analysis.