Real‐variable characterizations of local Hardy spaces on spaces of homogeneous type

Abstract

Let (X,d,μ) be a space of homogeneous type, with upper dimension μ, in the sense of R. R. Coifman and G. Weiss. Let η be the Hölder regularity index of wavelets constructed by P. Auscher and T. Hytönen. In this article, the authors introduce the local Hardy space h∗,pfalse(Xfalse) via local grand maximal functions and also characterize h∗,pfalse(Xfalse) via local radial maximal functions, local non‐tangential maximal functions, local atoms and local Littlewood–Paley functions. Furthermore, the authors establish the relationship between the global and the local Hardy spaces. Finally, the authors also obtain the finite atomic characterizations of h∗,pfalse(Xfalse). As an application, the authors give the dual spaces of h∗,pfalse(Xfalse) when p∈(ω/(ω+η),1), which further completes the result of G. Dafni and H. Yue on the dual space of h∗,1false(Xfalse). This article also answers the question of R. R. Coifman and G. Weiss on the nonnecessity of any additional geometric assumptions except the doubling condition for the radial maximal function characterization of Hnormalcw1false(Xfalse) when μ(X)<∞.

Maximal characterisation of local Hardy spaces on locally doubling manifolds

“…2 below for additional details). The atomic space of [16] coincides with the space of [18] in the case of manifolds with strongly bounded geometry (see Remark 2.7 below), and also with that of [3,9,21] in the case of doubling spaces. These works, however, do not address the issue of whether the local Hardy space admits characterisations analogous to (1.1) in a non-doubling setting.…”

“…A number of results in this direction are available in the case where the manifold is doubling. Indeed, an extensive theory of local Hardy spaces has been developed in the general context of doubling metric measure spaces (see, e.g., [3,8,9,21] and references therein), and includes both atomic and maximal characterisations. This theory is somewhat parallel to that of the "global" Hardy space H 1 à la Coifman-Weiss [2] on spaces of homogeneous type.…”

Besov and Triebel–Lizorkin spaces on spaces of homogeneous type with applications to boundedness of Calderón–Zygmund operators

“…Right after the Calderón reproducing formulae were established in [43], He et al [42] obtained a complete real-variable theory of atomic Hardy spaces on X with µ(X ) = ∞, which is equivalent to diam X = ∞ (see, for instance, Nakai and Yabuta [66,Lemma 5.1] or Auscher and Hytönen [3,Lemma 8.1]). Moreover, He et al [45] established some realvariable characterizations of local Hardy spaces on X without the assumption µ(X ) = ∞. We point out that, in both [42] and [45], He et al gave a complete answer to an open question asked by Coifman and Weiss [16, p. 642] on the radial maximal function characterization of Hardy spaces over X (see also [14, p. 5]).…”

“…Moreover, He et al [45] established some realvariable characterizations of local Hardy spaces on X without the assumption µ(X ) = ∞. We point out that, in both [42] and [45], He et al gave a complete answer to an open question asked by Coifman and Weiss [16, p. 642] on the radial maximal function characterization of Hardy spaces over X (see also [14, p. 5]). Later, Fu et al [22] obtained some real-variable characterizations of Musielak-Orlicz Hardy spaces on X .…”

“…At the end of this section, we establish the relationship between bmo(X ) and F 0 ∞,2 (X ). Let us begin with the notions of the 1-exp-IATI (see, for instance, [45,Definition 3.1]), bmo(X ) (see, for instance, [17, Defiinition 2.1]), and the local Hardy space h p (X ) (see, for instance, [45,Section 3]). Definition 6.16.…”