New Ball Campanato-Type Function Spaces and Their Applications

Zhang, Yangyang; Huang, Long; Yang, Dachun; Yuan, Wen

doi:10.1007/s12220-021-00822-x

Cited by 27 publications

(20 citation statements)

References 39 publications

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“…(iv) By Propositions 2.22(i) and 2.8, and Lemma 5.3, we conclude that all the assumptions of [119, Theorem 3.14] with X := Ė α, p q (R n ) are satisfied, and hence all the corresponding conclusions of [119,Theorem 3.14] with X := Ė α, p q (R n ) also hold true.…”

Section: Mixed-norm Herz-hardy Spacesmentioning

confidence: 64%

“…Recall that Sawano et al [88] established various real-variable characterizations of the Hardy space H X (R n ) associated with the ball quasi-Banach function space X, respectively, in terms of atoms, molecules, and Lusin area functions. Recently, the real-variable theory of function spaces associated with ball quasi-Banach function spaces is well developed; see, for instance, [59,57,8,116,51,119,96,88,120] for Hardy spaces associated with ball quasi-Banach function spaces, [48] for Erdélyi-Kober fractional integral operators on ball Banach function spaces, and [44,87,86] for other function spaces associated with ball quasi-Banach function spaces. Applying the existing results of H X (R n ), we establish a complete real-variable theory of Hardy spaces associated with mixed-norm Herz spaces.…”

Section: Introductionmentioning

confidence: 99%

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Mixed-Norm Herz Spaces and Their Applications in Related Hardy Spaces

Zhao¹,

Yang²,

Zhang³

2022

Preprint

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In this article, the authors introduce a class of mixed-norm Herz spaces, Ė α, p q (R n ), which is a natural generalization of mixed Lebesgue spaces and some special cases of which naturally appear in the study of the summability of Fourier transforms on mixed-norm Lebesgue spaces. The authors also give their dual spaces and obtain the Riesz-Thorin interpolation theorem on Ė α, p q (R n ). Applying these Riesz-Thorin interpolation theorem and using some ideas from the extrapolation theorem, the authors establish both the boundedness of the Hardy-Littlewood maximal operator and the Fefferman-Stein vector-valued maximal inequality on Ė α, p q (R n ). As applications, the authors develop various real-variable theory of Hardy spaces associated with Ė α, p q (R n ) by using the existing results of Hardy spaces associated with ball quasi-Banach function spaces. These results strongly depend on the duality of Ė α, p q (R n ) and the non-trivial constructions of auxiliary functions in the Riesz-Thorin interpolation theorem.

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Section: Mixed-norm Herz-hardy Spacesmentioning

confidence: 64%

Section: Introductionmentioning

confidence: 99%

Mixed-Norm Herz Spaces and Their Applications in Related Hardy Spaces

Zhao¹,

Yang²,

Zhang³

2022

Preprint

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“…From this, Theorem 4.19, and Theorem 4.23, we deduce that the dual space of H * Y (X) is the ball Campanatotype function space L Y,q,d (X) associated with Y(X). Remarkably, similarly to [39,75], to overcome the difficulty caused by the fact that the quasi-norm of Y(X) may not be concave, we consider the role as a whole of finite linear combinations of atoms instead of the role of a single atom in the construction of the ball Campanato-type function space; this makes perfect sense because the finite atomic Hardy space is dense in the corresponding Hardy space, and the most important thing is that both of them have equivalent quasi-norms.…”

Section: Introductionmentioning

confidence: 86%

“…Under much weaker assumptions on the Littlewood-Paley functions than the corresponding ones in [59,68], Chang et al [11] obtained various Littlewood-Paley function characterizations of H X (R n ). Recently, Zhang et al [76] and Wang et al [69] introduced weak Hardy-type spaces WH X (R n ) associated with X and developed a complete real-variable theory of these spaces; Yan et al [71] established the dual theory and obtained the intrinsic square function characterizations of H X (R n ); Zhang et al [75] introduced some new ball Campanato-type function space which proves the dual space of H X (R n ), and established its Carleson measure characterization. For more studies about ball quasi-Banach function spaces on R n , we refer the reader to [41,42,58,72].…”

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

Yan¹,

He²,

Yang³

et al. 2021

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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.

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“…Later, Wang et al [68] obtained the boundedness of both Calderón-Zygmund and pseudo-differential operators on both H X (R n ) and h X (R n ), the local version of H X (R n ); Chang et al [8] established various Littlewood-Paley function characterizations of H X (R n ). Very recently, Yan et al [72] obtained various intrinsic square function characterizations of H X (R n ); Zhang et al [75] introduced the ball Campanato-type function space L X,q,s,d (R n ) which proves the dual space of H X (R n ), and then established its Carleson measure characterization. We refer the reader to [62,63,67,68,69,70,71,76] for more studies on Hardy-type spaces associated with X.…”

Section: Introductionmentioning

confidence: 99%

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

Chen¹,

Jia²,

Yang³

2022

Preprint

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Let X be a ball quasi-Banach function space on R n and H X (R n ) the Hardy space associated with X, and let α ∈ (0, n) and β ∈ (1, ∞). In this article, assuming that the (powered) Hardy-Littlewood maximal operator satisfies the Fefferman-Stein vector-valued maximal inequality on X and is bounded on the associate space of X, the authors prove that the fractional integral I α can be extended to a bounded linear operator fromand only if there exists a positive constant C such that, for any ballX , where X β denotes the β-convexification of X. Moreover, under some different reasonable assumptions on both X and another ball quasi-Banach function space Y, the authors also consider the mapping property of I α from H X (R n ) to H Y (R n ) via using the extrapolation theorem. All these results have a wide range of applications. Particularly, when these are applied, respectively, to Morrey spaces, mixed-norm Lebesgue spaces, local generalized Herz spaces, and mixed-norm Herz spaces, all these results are new. The proofs of these theorems strongly depend on atomic and molecular characterizations of H X (R n ).

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New Ball Campanato-Type Function Spaces and Their Applications

Cited by 27 publications

References 39 publications

Mixed-Norm Herz Spaces and Their Applications in Related Hardy Spaces

Mixed-Norm Herz Spaces and Their Applications in Related Hardy Spaces

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

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

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