Boundedness of Fractional Integrals on Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

Chen, Yiqun; Jia, Hongchao; Yang, Dachun

doi:10.48550/arxiv.2206.06080

Cited by 3 publications

(4 citation statements)

References 57 publications

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“…Thus, by duality, the formal desired corresponding result of [10,Theorem 3.7] on the boundedness of fractional integrals on ball Campanato-type function spaces should be that…”

Section: Preliminariesmentioning

confidence: 99%

“…Obviously, due to the generality and the flexibility, more applications of these main results of this article are predictable. Moreover, we refer the reader to [10] for studies on fractional integrals on H X (R n ). To limit the length of this article, the boundedness of Calderón-Zygmund operators on L X,q,s,d (R n ) will be studied in the forthcoming article [11].…”

Section: Introductionmentioning

confidence: 99%

“…One key step is to show that, for any g ∈ (H X (R n )) * and any molecule M of H X (R n ), g, M = R n g(x)M(x) dx (see Lemma 3.30 below). To this end, we skillfully use the dual theorem on L X,q,s,d (R n ) obtained in [10], and the special atomic decomposition of molecules of H X (R n ) (see Proposition 3.28 below) to obtain the above desired conclusion.…”

Section: Introductionmentioning

confidence: 99%

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Boundedness of Fractional Integrals on Ball Campanato-Type Function Spaces

Chen¹,

Jia²,

Yang³

2022

Preprint

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Let X be a ball quasi-Banach function space on R n satisfying some mild assumptions and let α ∈ (0, n) and β ∈ (1, ∞). In this article, when α ∈ (0, 1), the authors first find a reasonable version I α of the fractional integral I α on the ball Campanato-type function space L X,q,s,d (R n ) with q ∈ [1, ∞), s ∈ Z n + , and d ∈ (0, ∞). Then the authors prove thatX , where X β denotes the β-convexification of X. Furthermore, the authors extend the range α ∈ (0, 1) in I α to the range α ∈ (0, n) and also obtain the corresponding boundedness in this case. Moreover, I α is proved to be the adjoint operator of I α . All these results have a wide range of applications. Particularly, even when they are applied, respectively, to Morrey spaces, mixed-norm Lebesgue spaces, local generalized Herz spaces, and mixed-norm Herz spaces, all the obtained results are new. The proofs of these results strongly depend on the dual theorem on L X,q,s,d (R n ) and also on the special atomic decomposition of molecules of H X (R n ) (the Hardy-type space associated with X) which proves the predual space of L X,q,s,d (R n ).

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“…Thus, by duality, the formal desired corresponding result of [10,Theorem 3.7] on the boundedness of fractional integrals on ball Campanato-type function spaces should be that…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Boundedness of Fractional Integrals on Ball Campanato-Type Function Spaces

Chen¹,

Jia²,

Yang³

2022

Preprint

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“…From [32, Theorem 1.2.20], we infer that K p,r ω,0 (R n ) is a ball quasi-Banach function space with p, r ∈ (0, ∞) and ω ∈ M(R + ) satisfying m 0 (ω) ∈ (− n p , ∞). However, it is worth pointing out that K p,r ω,0 (R n ) with p, r ∈ (0, ∞) and ω ∈ M (R + ) may not be a quasi-Banach function space (see, for instance, [9,Remark 4.13]).…”

Section: Local Generalized Herz Spacesmentioning

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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Fractional integral operators on Morrey spaces built on rearrangement-invariant quasi-Banach function spaces

2023

Positivity

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Boundedness of Fractional Integrals on Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

Cited by 3 publications

References 57 publications

Boundedness of Fractional Integrals on Ball Campanato-Type Function Spaces

Boundedness of Fractional Integrals on Ball Campanato-Type Function Spaces

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

Fractional integral operators on Morrey spaces built on rearrangement-invariant quasi-Banach function spaces

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