Commutators on Weighted Morrey Spaces on Spaces of Homogeneous Type

Gong, Ruming; Li, Ji; Pozzi, Elodie; Vempati, Manasa N.

doi:10.1515/agms-2020-0116

Cited by 10 publications

(5 citation statements)

References 31 publications

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“…This finishes the proof of Lemma 2.22. □ By borrowing some ideas from [23], we establish the following lemma which is regarded as a substitute of the reverse doubling condition and can be used to deal with integrals only in terms of measures of balls.…”

Section: Embedding Theorem For Ball Quasi-banach Function Spacesmentioning

confidence: 99%

“…By borrowing some ideas from [23], we establish the following lemma which is regarded as a substitute of the reverse doubling condition and can be used to deal with integrals only in terms of measures of balls. Lemma Let

{\mathcal {X}}

be a space of homogeneous type with

\mu ({\mathcal {X}})=\infty

c\in (1,\infty )

x_0\in {\mathcal {X}}

, and

r_0\in (0,\infty )

.…”

Section: Ball Quasi‐banach Function Spaces On Spaces Of Homogeneous Typementioning

confidence: 99%

See 1 more Smart Citation

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.

show abstract

Section: Embedding Theorem For Ball Quasi-banach Function Spacesmentioning

confidence: 99%

{\mathcal {X}}

be a space of homogeneous type with

\mu ({\mathcal {X}})=\infty

c\in (1,\infty )

x_0\in {\mathcal {X}}

, and

r_0\in (0,\infty )

.…”

Section: Ball Quasi‐banach Function Spaces On Spaces Of Homogeneous Typementioning

confidence: 99%

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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“…Since the kernel of is a standard kernel on , according to [6, Theorem 1.2], we can obtain that is bounded on and Thus, it suffices to show that …”

Section: Commutator Of Cauchy Type Integral For Bounded Strongly Pseu...mentioning

confidence: 99%

Boundedness and Compactness of Cauchy-Type Integral Commutator on Weighted Morrey Spaces

Gong¹,

Vempati²,

et al. 2022

J. Aust. Math. Soc.

Self Cite

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In this paper we study boundedness and compactness characterizations of the commutators of Cauchy type integrals on bounded strongly pseudoconvex domains D in $\mathbb C^{n}$ with boundaries $bD$ satisfying the minimum regularity condition $C^{2}$ , based on the recent results of Lanzani–Stein and Duong et al. We point out that in this setting the Cauchy type integral is the sum of the essential part which is a Calderón–Zygmund operator and a remainder which is no longer a Calderón–Zygmund operator. We show that the commutator is bounded on the weighted Morrey space $L_{v}^{p,\kappa }(bD)$ ( $v\in A_{p}, 1<p<\infty $ ) if and only if b is in the BMO space on $bD$ . Moreover, the commutator is compact on the weighted Morrey space $L_{v}^{p,\kappa }(bD)$ ( $v\in A_{p}, 1<p<\infty $ ) if and only if b is in the VMO space on $bD$ .

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“…By borrowing some ideas from [25], we establish the following lemma, which is regarded as a substitute of the reverse doubling condition and can be used to deal with integrals only in terms of measures of balls.…”

Section: 16)mentioning

confidence: 99%

Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Characterizations of Maximal Functions, Decompositions, and Dual Spaces

Yan¹,

He²,

Yang³

et al. 2021

Preprint

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Let (X, ρ, µ) be a space of homogeneous type in the sense of Coifman and Weiss, and Y(X) a ball quasi-Banach function space on X, which supports a Fefferman-Stein vectorvalued maximal inequality, and the boundedness of the powered Hardy-Littlewood maximal operator on its associate space. The authors first introduce the Hardy space H * Y (X), associated with Y(X), via the grand maximal function, and then establish its various real-variable characterizations, respectively, in terms of radial or non-tangential maximal functions, atoms or finite atoms, and molecules. As an application, the authors give the dual space of H * Y (X), which proves to be a ball Campanato-type function space associated with Y(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 article exist in that, to escape 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 expressed by dyadic reference points or dyadic cubes and, to overcome the difficulty caused by the lack of the good dense subset of H * Y (X), the authors further prove that Y(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.

show abstract

Commutators on Weighted Morrey Spaces on Spaces of Homogeneous Type

Cited by 10 publications

References 31 publications

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

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

Boundedness and Compactness of Cauchy-Type Integral Commutator on Weighted Morrey Spaces

Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Characterizations of Maximal Functions, Decompositions, and Dual Spaces

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