Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Littlewood—Paley Characterizations with Applications to Boundedness of Calderón—Zygmund Operators

Yan, Xian Jie; He, Zi Yi; Yang, Danni; Yuan, Wen

doi:10.1007/s10114-022-1573-9

Cited by 34 publications

(3 citation statements)

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“…We point out that the results obtained in this paper and [77] have a wide range of generality because 𝐻 * 𝑌 () includes various known Hardy spaces on , for instance, Hardy spaces in [33], Hardy-Lorentz spaces in [82], weighted Hardy spaces in [20], Orlicz-Hardy spaces in [20], and variable Hardy spaces on RD-spaces (namely, reverse doubling spaces) in [83]. Moreover, even when these results are applied to these concrete cases, some results are also new; obviously, more applications are expectable.…”

Section: Introductionsupporting

confidence: 60%

“…As an application, we also give the dual space of 𝐻 * 𝑌 (), which proves to be a ball Campanato-type function space associated with 𝑌(). These results can further be used to establish the Littlewood-Paley function characterizations of 𝐻 * 𝑌 () and the boundedness of Calderón-Zygmund operators on 𝐻 * 𝑌 (); to limit the length of this paper, they are presented in [77]. The major novelties of this paper exist in that, to escape both the reverse doubling condition of 𝜇 and the triangle inequality of 𝜌, we cleverly construct admissible sequences of balls and fully use the geometrical properties of  expressed by dyadic reference points and dyadic cubes and, to overcome the difficulty caused by the lack of the good dense subset of 𝑌(), we further prove that 𝑌() 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.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionsupporting

confidence: 60%

Section: Introductionmentioning

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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“…Then Sawano et al [23] introduced the Hardy type space H X (R n ). For more research works on Hardy spaces associated with ball quasi-Banach function spaces, see for instance [1,21,34,35].…”

Section: Introductionmentioning

confidence: 99%

Applications of Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

Wang

Yang

2020

Results Math

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Let X be a ball quasi-Banach function space satisfying some minor assumptions. In this article, the authors establish the characterizations of H X (R n ), the Hardy space associated with X, via the Littlewood-Paley g-functions and g * λ -functions. Moreover, the authors obtain the boundedness of Calderón-Zygmund operators on H X (R n ). For the local Hardytype space h X (R n ) associated with X, the authors also obtain the boundedness of S 0 1,0 (R n ) pseudo-differential operators on h X (R n ) via first establishing the atomic characterization of h X (R n ). Furthermore, the characterizations of h X (R n ) by means of local molecules and local Littlewood-Paley functions are also given. The results obtained in this article have a wide range of generality and can be applied to the classical Hardy space, the weighted Hardy space, the Herz-Hardy space, the Lorentz-Hardy space, the Morrey-Hardy space, the variable Hardy space, the Orlicz-slice Hardy space and their local versions. Some special cases of these applications are even new and, particularly, in the case of the variable Hardy space, the g * λ -function characterization obtained in this article improves the known results via widening the range of λ.As natural generalizations and substitutes of L p (R n ) with p ∈ (0, 1], a real-variable theory of classical Hardy spaces H p (R n ) was originally initiated by Stein and Weiss [51] and then systematically developed by Fefferman and Stein [17]. These celebrated articles [51] and [17] inspire many new ideas for the real-variable theory of function spaces. For instance, the characterizations of classical Hardy spaces reveal the important connections among various notions in harmonic analysis, such as harmonic functions, various maximal functions and various square functions.On another hand, due to the need from applications for more inclusive classes of function spaces than L p (R n ), many other function spaces are introduced; for instance, weighted Lebesgue spaces, Lorentz spaces, variable Lebesgue spaces, Orlicz spaces and Morrey spaces. These spaces and the Hardy-type spaces based on them have been investigated extensively. Associated with these spaces, an important concept about function spaces is the (quasi-)Banach function space, which is defined as follows (see, for instance, [6, Chapter 1] for more details).(iv) 1 E ∈ Y for any measurable set E ⊂ R n with finite measure. Here and thereafter, 1 E denotes the characteristic function of E.Moreover, a Banach space Y, consisting of measurable functions on R n , is called a Banach function space if it satisfies the above terms (i) through (iv) and(v) for any measurable set E ⊂ R n with finite measure, there exists a positive constant C (E) , depending on E, such that, for any f ∈ Y,One can show that Lebesgue spaces, Lorentz spaces, variable Lebesgue spaces and Orlicz spaces are (quasi-)Banach function spaces. However, weighted Lebesgue spaces, Herz spaces, Morrey spaces and Musielak-Orlicz spaces are not necessarily quasi-Banach function spaces (see, for instan...

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Bochner–Riesz Means on Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

Tan

Zhang

2023

Mediterr. J. Math.

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Let X (R n ) be a ball quasi-Banach function space on R n , W X (R n ) be the weak ball quasi-Banach function space on R n , H X (R n ) be the Hardy space associated with X (R n ) and W H X (R n ) be the weak Hardy space associated with X (R n ). In this paper, we obtain the boundedness of the Bochner-Riesz means and the maximal Bochner-Riesz means from H X (R n ) to W H X (R n ) or W X (R n ), which includes the critical case. Moreover, we apply these results to several examples of ball quasi-Banach function spaces, namely, weighted Lebesgue spaces, Herz spaces, Lorentz spaces, variable Lebesgue spaces and Morrey spaces. This shows that all the results obtained in this article are of wide applications, and more applications of these results are predictable.

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Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Littlewood—Paley Characterizations with Applications to Boundedness of Calderón—Zygmund Operators

Cited by 34 publications

References 65 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

Applications of Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

Bochner–Riesz Means on Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

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