Variable Hardy–Lorentz spaces

Jiao, Yong; Zuo, Yahui; Zhou, Dejian; Wu, Lian

doi:10.1002/mana.201700331

Cited by 46 publications

(30 citation statements)

References 47 publications

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“…From Corollary 2.22(i), it follows that, when X := L p (R n ) with p ∈ (0, 1], the range of the index λ in Corollary 2.22(i) coincides with the best-known classical case λ ∈ (2/p, ∞). [26,31,32,40]). We also point out that, for the space H p(·) (R n ), the range of the parameter λ, obtained in [26,31,32], is (1 + 2 min{2,p − } , ∞).…”

Section: Vanishes Weakly At Infinity and Smentioning

confidence: 99%

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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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Section: Vanishes Weakly At Infinity and Smentioning

confidence: 99%

“…[26,31,32,40]). We also point out that, for the space H p(·) (R n ), the range of the parameter λ, obtained in [26,31,32], is (1 + 2 min{2,p − } , ∞). Thus, our results improve those results when p − ∈ (0, 1] via widening the range (…”

Section: Vanishes Weakly At Infinity and Smentioning

confidence: 99%

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

Wang

Yang

2020

Results Math

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“…In recent years, the theory of function spaces with variable exponents has gained great interest, see, for example, [1][2][3][4][5][6][7][8][9]. e variable Lebesgue space is one of the generalizations of the classical L p (R n ) space, originally introduced by Orlicz [10] via replacing p by the variable exponent function p(·): R n ⟶ (0, ∞).…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Yan et al [8] introduced the variable weak Hardy space H p(·),∞ (R n ) by means of the radial grand maximal function and proved various characterizations including the atomic and molecular characterizations and investigated the boundedness of convolution δ-type and nonconvolution c-order Caldéron-Zygmund operators via the atomic characterization established in the same paper. Very recently, Jiao et al [4] investigated the variable Hardy-Lorentz space H p(·),q (R n ), constructed the atomic characterizations for this space, figured out its dual space, and proved the boundedness of singular integrals on H p(·),q (R n ).…”

Section: Introductionmentioning

confidence: 99%

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Intrinsic Square Function Characterizations of Variable Hardy–Lorentz Spaces

Saibi

2020

Journal of Function Spaces

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The aim of this paper is to establish the intrinsic square function characterizations in terms of the intrinsic Littlewood–Paley g-function, the intrinsic Lusin area function, and the intrinsic gλ∗-function of the variable Hardy–Lorentz space Hp⋅,qℝn, for p⋅ being a measurable function on ℝn satisfying 0<p−≔ess infx∈ℝnpx≤ess supx∈ℝnpx≕p+<∞ and the globally log-Hölder continuity condition and q∈0,∞, via its atomic and Littlewood–Paley function characterizations.

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Convergence of Vilenkin–Fourier series in variable Hardy spaces

Weisz

2022

Mathematische Nachrichten

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Let 𝑝(⋅) ∶ [0, 1) → (0, ∞) be a variable exponent function satisfying the log-Hölder condition and 0 < 𝑞 ≤ ∞. We introduce the variable Hardy and Hardy-Lorentz spaces 𝐻 𝑝(⋅) and 𝐻 𝑝(⋅),𝑞 containing Vilenkin martingales. We prove that the partial sums of the Vilenkin-Fourier series converge to the original function in the 𝐿 𝑝(⋅) -and 𝐿 𝑝(⋅),𝑞 -norm if 1 < 𝑝 − < ∞. We generalize this result for smaller 𝑝(⋅) as well. We show that the maximal operator of the Fejér means of the Vilenkin-Fourier series is bounded from 𝐻 𝑝(⋅) to 𝐿 𝑝(⋅) and from 𝐻 𝑝(⋅),𝑞 to 𝐿 𝑝(⋅),𝑞 if 1∕2 < 𝑝 − < ∞, 0 < 𝑞 ≤ ∞ and 1∕𝑝 − − 1∕𝑝 + < 1. This last condition is surprising because the corresponding results for Fourier series or Fourier transforms hold without this condition. This implies some norm and almost everywhere convergence results for the Fejér means of the Vilenkin-Fourier series.

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Variable Hardy–Lorentz spaces

Abstract: Let p(·) be a measurable function on double-struckRn with 0 Show more

Cited by 46 publications

References 47 publications

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

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

Intrinsic Square Function Characterizations of Variable Hardy–Lorentz Spaces

Convergence of Vilenkin–Fourier series in variable Hardy spaces

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