Localized Hardy spaces 𝐻¹ related to admissible functions on RD-spaces and applications to Schrödinger operators

Abstract

Abstract. Let X be an RD-space, which means that X is a space of homogenous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in X . In this paper, the authors first introduce the notion of admissible functions ρ and then develop a theory of localized Hardy spaces H 1 ρ (X ) associated with ρ, which includes several maximal function characterizations of H 1 ρ (X ), the relations between H 1 ρ (X ) and the classical Hardy space H 1 (X ) via constructing a kernel function related to ρ, the atomic decomposition characterization of H 1 ρ (X ), and the boundedness of certain localized singular integrals on H 1 ρ (X ) via a finite atomic decomposition characterization of some dense subspace of H 1 ρ (X ). This theory has a wide range of applications. Even when this theory is applied, respectively, to the Schrödinger operator or the degenerate Schrödinger operator on R n , or to the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups, some new results are also obtained. The Schrödinger operators considered here are associated with nonnegative potentials satisfying the reverse Hölder inequality.

Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type

“…We shall leave these for a future project. Thirdly our approach can be adapted to settings with reverse doubling to give maximal function characterizations in terms of certain 'approximations of the identity', extending the results in [42] for p = 1 to 0 < p ≤ 1. See Remark 5.8.…”

“…We wish to point out that the atomic space in (5) is a modification of the atomic spaces of Coifman and Weiss -see Definition 2.9. The spaces in (5) and their identification were originally studied in [15,16,19] for X = R n , while variations have since been considered in say [17,18,30,42].…”

Calderón‐Zygmund operators with non‐diagonal singularity

“…Therefore, the classical Calderón-Zygmund theory is a powerful tool in many aspects of harmonic analysis and partial differential equations [6,7,10,11,13,18,[23][24][25][26][27][28]. And it has been widely studied in various directions, e.g., see [1-5, 8, 9, 12] and references therein.…”

Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrüdinger operators

“…( [21], [30], [31] [33], [32]). Observe that { T t 2 } t>0 satisfies that for all t ∈ (0, ∞), T t 2 (1) = 1 (see, e.g., [30]).…”

“…It is well known that the dual space of the Hardy space H p (R d ) with p ∈ (0, 1) is the Morrey-Campanato space E 1/p−1,1 (R d ). Notice that Morrey-Campanato spaces on R d are essentially related to the Laplacian Δ, where Δ ≡ On the other hand, there exists an increasing interest in the study of Schrödinger operators on R d and the sub-Laplace Schrödinger operators on connected and simply connected nilpotent Lie groups with nonnegative potentials satisfying the reverse Hölder inequality (see, e.g., [10], [34], [25], [18], [8], [7], [19], [33], [16]). Let L ≡ −Δ + V be the Schrödinger operator on R d , where the potential V is a nonnegative locally integrable function.…”