A Complete Real-Variable Theory of Hardy Spaces on Spaces of Homogeneous Type

He, Ziyi; Han, Yongsheng; Ji, Li; Liu, Liguang; Yang, Dachun; Yuan, Wen

doi:10.1007/s00041-018-09652-y

Cited by 47 publications

(138 citation statements)

References 69 publications

(176 reference statements)

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“…It is these Calderón reproducing formulae built in the present article that impel us to establish, without any additional geometrical condition, various real-variable characterizations of Hardy spaces on spaces homogeneous type, which completely answers the question asked by Coifman and Weiss [11, p. 642]. Due to the limited length of this article, the latter part is presented in [32].…”

Section: Introductionsupporting

confidence: 56%

New Calderón reproducing formulae with exponential decay on spaces of homogeneous type

Liu

Yang

et al. 2019

Sci. China Math.

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Assume that (X, d, µ) is a space of homogeneous type in the sense of Coifman and Weiss. In this article, motivated by the breakthrough work of P. Auscher and T. Hytönen on orthonormal bases of regular wavelets on spaces of homogeneous type, the authors introduce a new kind of approximations of the identity with exponential decay (for short, exp-ATI). Via such an exp-ATI, motivated by another creative idea of Y. Han et al. to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, the authors establish the homogeneous continuous/discrete Calderón reproducing formulae on (X, d, µ), as well as their inhomogeneous counterparts. The novelty of this article exists in that d is only assumed to be a quasi-metric and the underlying measure µ a doubling measure, not necessary to satisfy the reverse doubling condition. It is well known that Calderón reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.2010 Mathematics Subject Classification. Primary 42C40; Secondary 42B20, 42B25, 30L99. Key words and phrases. space of homogeneous type, Calderón reproducing formula, approximation of the identity, wavelet, space of test functions, distribution.

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Section: Introductionsupporting

confidence: 56%

New Calderón reproducing formulae with exponential decay on spaces of homogeneous type

Liu

Yang

et al. 2019

Sci. China Math.

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“…Then, in Subsection 5.2, we obtain a version of Calderón-Zygmund decompositions for H * ,ϕ (X ) (see Proposition 5.8 below), which is a generalization of the corresponding results on H p (X ) in [27,Proposition 4.9]. In Subsection 5.3, via this Calderón-Zygmund decompositions for H * ,ϕ (X ), a technical lemma on the density of L q ϕ(•,1) (X ) ∩H * ,ϕ (X ) in H * ,ϕ (X ) and some arguments similar to those used in the proof of [27,Theorem 4.2], we prove that H * ,ϕ (X ) ⊂ H ϕ,q at (X ), namely, we establish the atomic decomposition of H * ,ϕ (X ).…”

Section: Introductionmentioning

confidence: 85%

“…More recently, He et al [28] constructed a corresponding wavelet reproducing formulae without having recourse to the reverse doubling condition (1.5). Based on these wavelet reproducing formulae, He et al [27] further established a complete real-variable theory of Hardy spaces H p (X ) on spaces of homogeneous type without having recourse to the reverse doubling condition (1.5). On another hand, Hou et al [31] investigated Musielak-Orlicz BMO-type spaces associated with generalized approximations to the identity on spaces of homogeneous type.…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by [27] and the useful space H log (X ), one naturally expects to develop a complete real-variable theory of Musielak-Orlicz Hardy spaces, including the above useful space H log (X ). In this article, we positively answer this question on (X , d, µ).…”

Section: Introductionmentioning

confidence: 99%

“…In Section 4, we first recall the important Calderón reproducing formulae established in [27] and then introduce Musielak-Orlicz Hardy spaces, H +,ϕ (X ), H ϕ θ (X ) and H * ,ϕ (X ) defined, respectively, via radial, nontangential and grand maximal functions. In Subsection 4.2, by the pointwise inequalities for those maximal functions and some ideas from the proof of [27,Theorem 3.5], we show that H +,ϕ (X ), H ϕ θ (X ) and H * ,ϕ (X ) are mutually identical with equivalent quasi-norms (see Theorem 4.12 below).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Real-variable characterizations of Musielak–Orlicz Hardy spaces on spaces of homogeneous type

Fu¹,

Ma²,

Yang³

2020

Ann. Acad. Sci. Fenn. Math.

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Let (X , d, µ) be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors establish a complete real-variable theory of Musielak-Orlicz Hardy spaces on (X , d, µ). To be precise, the authors first introduce the atomic Musielak-Orlicz Hardy space H ϕ at (X) and then establish its various maximal function characterizations. The authors also investigate the Littlewood-Paley characterizations of H ϕ at (X) via Lusin area functions, Littlewood-Paley g-functions and Littlewood-Paley g * λ-functions. The authors further obtain the finite atomic characterization of H ϕ at (X) and its improved version in case q < ∞, and their applications to criteria of the boundedness of sublinear operators from H ϕ at (X) to a quasi-Banach space, which are also applied to the boundedness of Calderón-Zygmund operators. Moreover, the authors find the dual space of H ϕ at (X), namely, the Musielak-Orlicz BMO space BMO ϕ (X), present its several equivalent characterizations, and apply it to establish a new characterization of the set of pointwise multipliers for the space BMO(X). The main novelty of this article is that, throughout the article, except the last section, µ is not assumed to satisfy the reverse doubling condition.

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

Yang

Yuan

2021

Mathematische Nachrichten

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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)<∞.

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A Complete Real-Variable Theory of Hardy Spaces on Spaces of Homogeneous Type

Abstract: Suppose that (X, d, µ) is a space of homogeneous type, with upper dimension

Cited by 47 publications

References 69 publications

New Calderón reproducing formulae with exponential decay on spaces of homogeneous type

New Calderón reproducing formulae with exponential decay on spaces of homogeneous type

Real-variable characterizations of Musielak–Orlicz Hardy spaces on spaces of homogeneous type

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

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