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

Zhao, Yongxiang; Yang, Dachun; Zhang, Yangyang

doi:10.48550/arxiv.2204.12019

Cited by 3 publications

(10 citation statements)

References 88 publications

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“…Then, by [77,Lemma 5.3(ii)] and its proof, we conclude that Assumption 2.10 also holds true with X := Ė α, p q (R n ). Thus, all the assumptions of Theorem 3.7 with X := Ė α, p q (R n ) hold true.…”

Section: Mixed Herz Spacesmentioning

confidence: 65%

“…Recall that a special case of the mixed Herz space was originally introduced by Huang et al [35] to study the Lebesgue points of functions in mixed-norm Lebesgue spaces and, later, Zhao et al [77] generalized it to the above case. Also, both the dual theorem and the Riesz-Thorin interpolation theorem on the above mixed Herz space have been fully studied in [77].…”

Section: Mixed Herz Spacesmentioning

confidence: 99%

“…The following theorem is a corollary of Theorem 3.7. In what follows, the mixed Herz-Hardy space H Ė α, p q (R n ) is defined as in Definition 2.7 with X := Ė α, p q (R n ) (see [77,Definition 5.2]).…”

Section: Mixed Herz Spacesmentioning

confidence: 99%

See 2 more Smart Citations

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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Section: Mixed Herz Spacesmentioning

confidence: 65%

Section: Mixed Herz Spacesmentioning

confidence: 99%

See 1 more Smart Citation

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

Chen¹,

Jia²,

Yang³

2022

Preprint

Self Cite

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“…, α n ) ∈ R n . By [78,Proposition 2.22], we conclude that Ė α, p q (R n ) is a ball quasi-Banach space if and only if, for any i ∈ {1, . .…”

Section: Morrey Spacesmentioning

confidence: 92%

“…Recall that, to study the Lebesgue points of functions in mixed-norm Lebesgue spaces, Huang et al [33] introduced a special case of the mixed Herz space and, later, Zhao et al [78] generalized it to the above case. Also, both the dual theorem and the Riesz-Thorin interpolation theorem on the above mixed Herz space have been fully studied in [78].…”

Section: Mixed Herz Spacesmentioning

confidence: 99%

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

Cited by 3 publications

References 88 publications

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

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

Boundedness of Fractional Integrals on Ball Campanato-Type Function Spaces

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

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