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

Wang, Fan; Han, Yongsheng; He, Ziyi; Yang, Dachun

doi:10.4064/dm821-4-2021

Cited by 11 publications

(16 citation statements)

References 78 publications

(148 reference statements)

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“…The proofs of the following two lemmas are similar, respectively, to those of [65,Lemma 3.5] and [32,Lemma 5.3]; we omit the details here.…”

Section: }} and The Following Integral Condition That For Anymentioning

confidence: 99%

“…It was proved in [65] that these spaces are independent of the choices of β and γ as in (3.1), and exp-ATIs (see [65,Remark 3.13]), which makes Definition 3.1 well defined. Now, we state our main results in this section, namely, the wavelet characterization of Ḃs p,q (X) and Ḟs p,q (X).…”

Section: Wavelet Characterization Of Besov and Triebel-lizorkin Spacesmentioning

confidence: 99%

“…Motivated by these previous works, on one hand, Wang et al [65] introduced Besov and Triebel-Lizorkin spaces on a given space X of homogeneous type via the exp-(I)ATI, and also obtained the boundedness of Calderón-Zygmund operators on these function spaces, and Wang et al [66] further established the difference characterization of these spaces. On the other hand, Han et al [27] introduced another kind of Besov and Triebel-Lizorkin spaces on X by using the wavelet system introduced in [2], and gave a necessary and sufficient condition of the embedding theorem of Besov and Triebel-Lizorkin spaces.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, Han et al [27] introduced another kind of Besov and Triebel-Lizorkin spaces on X by using the wavelet system introduced in [2], and gave a necessary and sufficient condition of the embedding theorem of Besov and Triebel-Lizorkin spaces. Comparing the results in [27] with those in [65], the following question naturally arises. Question 1.1.…”

Section: Introductionmentioning

confidence: 99%

“…Question 1.1. Do these two kinds of Besov and Triebel-Lizorkin spaces on a given space X of homogeneous type introduced, respectively, in [65] and [27], coincide?…”

Section: Introductionmentioning

confidence: 99%

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Wavelet Characterization of Besov and Triebel--Lizorkin Spaces on Spaces of Homogeneous Type and Its Applications

Wang

Yang

et al. 2021

Preprint

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In this article, the authors establish the wavelet characterization of Besov and Triebel-Lizorkin spaces on a given space (X, d, µ) of homogeneous type in the sense of Coifman and Weiss. Moreover, the authors introduce almost diagonal operators on Besov and Triebel-Lizorkin sequence spaces on X, and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal operators, the authors obtain the molecular characterization of Besov and Triebel-Lizorkin spaces. Applying this molecular characterization, the authors further establish the Littlewood-Paley characterizations of Triebel-Lizorkin spaces on X. The main novelty of this article is that all these results get rid of their dependence on the reverse doubling property of µ and also the triangle inequality of d, by fully using the geometrical property of X expressed via its equipped quasi-metric d, dyadic reference points, dyadic cubes, and wavelets.

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“…The proofs of the following two lemmas are similar, respectively, to those of [65,Lemma 3.5] and [32,Lemma 5.3]; we omit the details here.…”

Section: }} and The Following Integral Condition That For Anymentioning

confidence: 99%

Section: Wavelet Characterization Of Besov and Triebel-lizorkin Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Question 1.1. Do these two kinds of Besov and Triebel-Lizorkin spaces on a given space X of homogeneous type introduced, respectively, in [65] and [27], coincide?…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Wavelet Characterization of Besov and Triebel--Lizorkin Spaces on Spaces of Homogeneous Type and Its Applications

Wang

Yang

et al. 2021

Preprint

Self Cite

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Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces

Yan

Yang

et al. 2023

Mathematische Nachrichten

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Let false(scriptX,ρ,μfalse)$({\mathcal {X}},\rho ,\mu )$ be a space of homogeneous type in the sense of Coifman and Weiss, and let Yfalse(scriptXfalse)$Y({\mathcal {X}})$ be a ball quasi‐Banach function space on X${\mathcal {X}}$, which supports both a Fefferman–Stein vector‐valued maximal inequality and the boundedness of the powered Hardy–Littlewood maximal operator on its associate space. The authors first introduce the Hardy space HY∗(X)$H_{Y}^*({\mathcal {X}})$, associated with Yfalse(scriptXfalse)$Y({\mathcal {X}})$, via the grand maximal function and then establish its various real‐variable characterizations, respectively, in terms of radial or nontangential maximal functions, atoms or finite atoms, and molecules. As an application, the authors give the dual space of HY∗(X)$H_{Y}^*({\mathcal {X}})$, which proves to be a ball Campanato‐type function space associated with Yfalse(scriptXfalse)$Y({\mathcal {X}})$. All these results have a wide range of generality and, particularly, even when they are applied to variable Hardy spaces, the obtained results are also new. The major novelties of this paper exist in that, to escape both the reverse doubling condition of μ and the triangle inequality of ρ, the authors cleverly construct admissible sequences of balls and fully use the geometrical properties of X${\mathcal {X}}$ expressed by dyadic reference points or dyadic cubes and, to overcome the difficulty caused by the lack of the good dense subset of HY∗(X)$H_{Y}^*({\mathcal {X}})$, the authors further prove that Yfalse(scriptXfalse)$Y({\mathcal {X}})$ can be embedded into the weighted Lebesgue space with certain special weight and then can fully use the known results of the weighted Lebesgue space.

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Geometric Characterizations of Embedding Theorems: For Sobolev, Besov, and Triebel–Lizorkin Spaces on Spaces of Homogeneous Type—via Orthonormal Wavelets

Han

et al. 2020

J Geom Anal

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

Cited by 11 publications

References 78 publications

Wavelet Characterization of Besov and Triebel--Lizorkin Spaces on Spaces of Homogeneous Type and Its Applications

Wavelet Characterization of Besov and Triebel--Lizorkin Spaces on Spaces of Homogeneous Type and Its Applications

Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces

Geometric Characterizations of Embedding Theorems: For Sobolev, Besov, and Triebel–Lizorkin Spaces on Spaces of Homogeneous Type—via Orthonormal Wavelets

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