A Local Version of Hardy-type Spaces Associated with Ball Quasi-Banach Spaces and Non-negative Self-adjoint Operators on Spaces of Homogeneous Type and Their Applications

TL;DR

This work develops a local Hardy-space theory $h_L^X({\mathbb X})$ on spaces of homogeneous type for a ball quasi-Banach space $X$ and a non-negative self-adjoint operator $L$ whose heat kernel satisfies Gaussian bounds. The authors introduce local area and tent-space tools and prove a molecular characterization of $h_L^X({\mathbb X})$ via $(X,2,M,\varepsilon)$-molecules, establishing inclusion relations with the global spaces $H_L^X({\mathbb X})$ or $H^X_{L+I}({\mathbb X})$ under spectral assumptions like $\inf\sigma(L)>0$. The framework is then specialized to concrete function spaces, including local Orlicz-Hardy spaces, local variable Hardy spaces, and local mixed-norm Hardy spaces, yielding explicit identifications and equivalences in each case. These results extend Goldberg's local theory to the non-Euclidean, operator-driven setting, offering a versatile toolkit for harmonic analysis and PDEs on spaces of homogeneous type with broad function-space generality. The combination of local tent-space techniques, molecular decompositions, and operator-theoretic Hardy spaces broadens applicability to diverse contexts, enabling sharper $L^p$-type bounds for singular integrals and related operators in non-Euclidean geometries.

Abstract

Let $(\mathbb{X},\,d,\,μ)$ be a space of homogeneous type in the sense of Coifman and Weiss, $X$ be a ball quasi-Banach function space on $\mathbb{X}$, $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{X})$, and assume that, for all $t>0$, the semigroup $e^{-tL}$ has an integral representation whose kernel satisfies a Gaussian upper bound condition. In this paper, we first study a local version of Hardy space $h^{X}_L(\mathbb{X})$ associated with ball quasi-Banach space $X$ and non-negative self-adjoint operator $L$, which is an extension of Goldberg's result [Duke Math. J. {\bf46} (1979), no. 1, 27-42; MR0523600]. Even in the case of Euclidean space (that is, $\mathbb{X}=\mathbb{R}^d$), all of these results are still new.

Theorems & Definitions (25)

• Lemma 2.1: ghl09
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• Definition 2.5
• Definition 2.8
• Lemma 2.9
• Definition 2.10
• Lemma 2.11
• Theorem 3.2