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

Abstract

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 Calder\'on 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 Calder\'on--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.