Orlicz-Lorentz Hardy martingale spaces

Hao, Zhiwei; Li, Libo

doi:10.1016/j.jmaa.2019.123520

Cited by 28 publications

(11 citation statements)

References 28 publications

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“…16 If we consider the special case (t) = t p for 0 < p < ∞ with the notations above, then we obtain the definition of H M p,q,b and H S p,q,b , respectively; see Ho [28]. If b ≡ 1, we obtain the definition of H M ,q and H S ,q , respectively; see Hao and Li [23]. In addition, if (t) = t p for 0 < p < ∞, then we obtain the definition of H M p,q and H S p,q , respectively; see Weisz [56] and Jiao et al [30,33].…”

Section: Martingale Hardy Orlicz-lorentz-karamata Spacesmentioning

confidence: 99%

“…In 2017, Liu and Zhou [37] investigated the boundedness of fractional integral operators on Lorentz-Karamata martingale spaces. Motivated by these results, the authors [25,26] introduced a class of Orlicz-Lorentz-Karamata spaces, which are much wider than the Lorentz-Karamata spaces or the Orlicz-Lorentz spaces and other important function spaces, and then developed a theory of the martingale Hardy spaces in the new framework. The method of atomic decomposition as one of results about martingale Hardy Orlicz-Lorentz-Karamata spaces comes from [33], which is a very useful tool in martingale theory.…”

Section: Introductionmentioning

confidence: 99%

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Applications of martingale Hardy Orlicz–Lorentz–Karamata theory in Fourier analysis

Hao,

Li,

Weisz

2024

Banach J. Math. Anal.

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In this article, we discuss the applications of martingale Hardy Orlicz–Lorentz–Karamata spaces in Fourier analysis. More precisely, we show that the partial sums of the Walsh–Fourier series converge to the function in norm if $$f\in L_{\Phi ,q,b}$$ f ∈ L Φ , q , b with $$1<p_-\le p_+<\infty $$ 1 < p - ≤ p + < ∞ . The equivalence of maximal operators on martingale Hardy Orlicz–Lorentz–Karamata spaces is presented. The Fejér summability method is also studied and it is proved that the maximal Fejér operator is bounded from martingale Hardy Orlicz–Lorentz–Karamata spaces to Orlicz–Lorentz–Karamata spaces. As a consequence, we obtain conclusions about almost everywhere and norm convergence of Fejér means.

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Section: Martingale Hardy Orlicz-lorentz-karamata Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Applications of martingale Hardy Orlicz–Lorentz–Karamata theory in Fourier analysis

Hao,

Li,

Weisz

2024

Banach J. Math. Anal.

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“…Sadasue [40] proved the boundedness of fractional integrals on martingale Hardy spaces for

0&lt;p\le 1

. Since then, the notion of fractional integrals has been extended to more general martingale spaces, for instance, martingale Morrey spaces [39], martingale Hardy–Lorentz spaces [27], martingale Orlicz–Lorentz Hardy spaces [20], martingale Hardy spaces with variable exponents [19]. Very recently, Zeng [49] extended the boundedness of fractional integral operator to variable martingale Hardy spaces

\mathcal {H}_{p(\cdot )}^M

and

\mathcal {H}_{p(\cdot ),q}^M

.…”

Section: Boundedness Of Fractional Integrals On Variable Hardy–lorent...mentioning

confidence: 99%

Variable Hardy–Lorentz martingale spaces Hp(·),q(·)$\mathcal {H}_{p(\cdot ),q(\cdot )}$

2023

Mathematische Nachrichten

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Let be a complete probability space and let be two variable exponents. In this paper, we investigate several new variable Hardy–Lorentz martingale spaces associated with the variable Lorentz spaces , which are defined by nonincreasing rearrangement functions. To be precise, we first formulate the atomic decomposition theorems for these Hardy–Lorentz martingale spaces, and then establish martingale inequalities among them with the help of the boundedness of a σ‐sublinear operator. Furthermore, we prove the boundedness of fractional integrals on variable Hardy–Lorentz martingale spaces .

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“…In addition, the weak quasi-Banach function lattice WX has generalized a number of classical weak spaces including the weak variable space, weak Orlicz space, weak Morrey space, and weak Musielak-Orlicz space; we refer readers to [13,24,25,42,52] for definitions of these weak spaces.…”

Section: Rearrangement-invariant Quasi-banach Function Spacementioning

confidence: 99%

Dual spaces for weak martingale Hardy spaces associated with rearrangement-invariant spaces

Silas

Xie

2022

Preprint

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Given a probability space $(\Omega,\mathcal{F},\mathbb P)$ and a rearrangement-invariant quasi-Banach function space $X$, the authors of this article first prove the $\alpha$-atomic ($\alpha\in [1,\infty)$) characterization of weak martingale Hardy spaces $WH_X(\Omega)$ associated with $X$ via simple atoms. The authors then introduce the generalized weak martingale ${\rm BMO}$ spaces which proves to be the dual spaces of $WH_X(\Omega)$. Consequently, the authors derive a new John--Nirenberg theorem for these weak martingale ${\rm BMO}$ spaces. Finally, the authors apply these results to the generalized grand Lebesgue space and the weighted Lorentz space. Even in these special cases, the results obtained in this article are totally new.

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Orlicz-Lorentz Hardy martingale spaces

Cited by 28 publications

References 28 publications

Applications of martingale Hardy Orlicz–Lorentz–Karamata theory in Fourier analysis

Applications of martingale Hardy Orlicz–Lorentz–Karamata theory in Fourier analysis

Variable Hardy–Lorentz martingale spaces Hp(·),q(·)$\mathcal {H}_{p(\cdot ),q(\cdot )}$

Dual spaces for weak martingale Hardy spaces associated with rearrangement-invariant spaces

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