Martingale Morrey-Hardy and Campanato-Hardy Spaces

Nakai, Eiichi; Sadasue, Gaku; Sawano, Yoshihiro

doi:10.1155/2013/690258

Cited by 24 publications

(7 citation statements)

References 23 publications

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“…Moreover, martingale Lorentz-Karamata Hardy spaces and multi-parameter martingale Hardy spaces were investigated by Ho [29], Jiao et al [36] and Weisz [75]. Variable martingale Hardy spaces, martingale Morrey Hardy spaces, martingale BLO spaces, martingale Besov spaces and Tribel-Lizorkin spaces were studied by Nakai et al [56,55,57,52,66]. Martingale Musielak-Orlicz Hardy spaces were dealt with by Xie at al.…”

Section: Introductionmentioning

confidence: 99%

New martingale inequalities and applications to Fourier analysis

et al. 2019

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Let (Ω, F , P) be a probability space and ϕ : Ω × [0, ∞) → [0, ∞) be a Musielak-Orlicz function. In this article, the authors prove that the Doob maximal operator is bounded on the Musielak-Orlicz space L ϕ (Ω). Using this and extrapolation method, the authors then establish a Fefferman-Stein vector-valued Doob maximal inequality on L ϕ (Ω). As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for L ϕ (Ω), which are new even in weighted Orlicz spaces. The authors then establish the atomic characterizations of martingale Musielak-Orlicz Hardy spaces H s ϕ (Ω), P ϕ (Ω), Q ϕ (Ω), H S ϕ (Ω) and H M ϕ (Ω). From these atomic characterizations, the authors further deduce some martingale inequalities between different martingale Musielak-Orlicz Hardy spaces, which essentially improve the corresponding results in Orlicz space case and are also new even in weighted Orlicz spaces. By establishing the Davis decomposition on H S ϕ (Ω) and H M ϕ (Ω), the authors obtain the Burkholder-Davis-Gundy inequality associated with Musielak-Orlicz functions. Finally, using the previous martingale inequalities, the authors prove that the maximal Fejér operator is bounded from H ϕ [0, 1) to L ϕ [0, 1), which further implies some convergence results of the Fejér means; these results are new even for the weighted Hardy spaces.

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Section: Introductionmentioning

confidence: 99%

New martingale inequalities and applications to Fourier analysis

et al. 2019

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“…Recently, various martingale Hardy spaces were investigated; see, for example, Weisz [26,24,23], Ho [7,8], Nakai et al [19,20,21], Sadasue [22] and Jiao et al [11,27] for various different martingale Hardy spaces and their applications. Moreover, the theory of weak martingale Hardy spaces has also been developed rapidly.…”

Section: Introductionmentioning

confidence: 99%

Atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces and their applications

Xie¹,

Yang²

2019

Banach J. Math. Anal.

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Let (Ω, F , P) be a probability space and ϕ : Ω × [0, ∞) → [0, ∞) be a Musielak-Orlicz function. In this article, the authors establish the atomic characterizations of weak martingale Musielak-Orlicz Hardy spaces WH s ϕ (Ω), WH M ϕ (Ω), WH S ϕ (Ω), WP ϕ (Ω) and WQ ϕ (Ω). Using these atomic characterizations, the authors then obtain the boundedness of sublinear operators from weak martingale Musielak-Orlicz Hardy spaces to weak Musielak-Orlicz spaces, and some martingale inequalities which further clarify the relationships among WH s ϕ (Ω), WH M ϕ (Ω), WH S ϕ (Ω), WP ϕ (Ω) and WQ ϕ (Ω). All these results improve and generalize the corresponding results on weak martingale Orlicz-Hardy spaces. Moreover, the authors also improve all the known results on weak martingale Musielak-Orlicz Hardy spaces. In particular, both the boundedness of sublinear operators and the martingale inequalities, for the weak weighted martingale Hardy spaces as well as for the weak weighted martingale Orlicz-Hardy spaces, are new. IntroductionAs is well known, the classical weak Hardy spaces naturally appear when studying the boundedness of operators in critical cases. Indeed, Fefferman and Soria [6] originally introduced the weak Hardy space W H 1 (R n ) and proved in [6, Theorem 5] that some Calderón-Zygmund operators are bounded from W H 1 (R n ) to weak Lebesgue spaces W L 1 (R n ). It should also point out that Fefferman et al. [5] proved that the weak Hardy spaces are the intermediate spaces of Hardy spaces in the real interpolation method. Recently, various martingale Hardy spaces were investigated; see, for example, Weisz [26, 24, 23], Ho [7, 8], Nakai et al. [19, 20, 21], Sadasue [22] and Jiao et al. [11, 27] for various different martingale Hardy spaces and their applications. Moreover, the theory of weak martingale Hardy spaces has also been developed rapidly. The weak Hardy spaces consisting of Vilenkin martingales were originally studied by Weisz [25] and then fully generalized by Hou and Ren [9]. Inspired by these, Jiao et al. [13, 12] and Liu et al. [17, 16] investigated the weak martingale Orlicz-Hardy spaces associated with concave functions. Zhou et al. [31] introduced the weak martingale 2010 Mathematics Subject Classification. Primary 60G42; Secondary 60G46, 42B25, 42B35.

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“…The study of martingale Hardy spaces generated by Orlicz spaces is given in [6]. The family of martingale function spaces had been generalized to Lebesgue spaces with variable exponents [7][8][9] and Morrey spaces [10][11]. For martingales in variable exponent spaces we use more some different methods than classical martingale spaces.…”

Section: Introductionmentioning

confidence: 99%

Weighted Variable Exponent Lebesgue Spaces on a Probability Space

AYDIN,

AYDIN

2023

J. Sci. Arts

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In this paper, we introduce the weighted variable exponent Lebesgue spaces defined on a probability space and give some information about the martingale theory of these spaces. We first prove several basic inequalities for expectation operators and obtain several norm convergence conditions for martingales in weighted variable exponent Lebesgue spaces. We discuss the Hölder inequality and embedding properties in these spaces. Finally, under some conditions we investigate Doob’s maximal function.

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Martingale Morrey-Hardy and Campanato-Hardy Spaces

Cited by 24 publications

References 23 publications

New martingale inequalities and applications to Fourier analysis

New martingale inequalities and applications to Fourier analysis

Atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces and their applications

Weighted Variable Exponent Lebesgue Spaces on a Probability Space

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