Martingales and function spaces

Kikuchi, Masato

doi:10.1090/suga/431

Cited by 5 publications

(3 citation statements)

References 46 publications

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“…The following lemma gives the boundedness of the Doob maximal operator; see, for instance, [26,Corollary 3.6] and [36,Theorem B]. We refer the reader to [16,33,34,35] for martingale inequalities on Banach function space.…”

Section: Weak Martingale Hardy-type Spacesmentioning

confidence: 99%

Dual Spaces for Weak Martingale Hardy Spaces Associated with Rearrangement-Invariant Spaces

Quan,

Silas,

Xie

2023

Potential Anal

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Given a probability space (Ω, F , P) and a rearrangement-invariant quasi-Banach function space X, the authors of this article first prove the α-atomic (α ∈ [1, ∞)) characterization of weak martingale Hardy spaces WH X (Ω) associated with X via simple atoms. The authors then introduce the generalized weak martingale BMO spaces which proves to be the dual spaces of WH X (Ω). Consequently, the authors derive a new John-Nirenberg theorem for these weak martingale 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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Section: Weak Martingale Hardy-type Spacesmentioning

confidence: 99%

Dual Spaces for Weak Martingale Hardy Spaces Associated with Rearrangement-Invariant Spaces

Quan,

Silas,

Xie

2023

Potential Anal

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show abstract

“…The following lemma gives the boundedness of the Doob maximal operator; see, for instance, [22,Corollary 3.6] and [32,Theorem B]. We refer the reader to [15,29,30,31] for martingale inequalities on Banach function space. Lemma 2.10.…”

Section: Weak Martingale Hardy-type Spacesmentioning

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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“…Indeed, L pÞ ð½0 , 1ÞÞ is a rearrangement-invariant Banach function space, L pÞ ð½0, 1ÞÞ ≠ L 1 ð½0, 1ÞÞ, but is not of absolutely continuous norm from [20]. According to Theorem 3.3 in [25], there exists a martingale f = ðf n Þ n≥0 such that it does not converge in L pÞ ð½0, 1ÞÞ.…”

Section: Preliminariesmentioning

confidence: 99%

The Boundedness of Doob’s Maximal and Fractional Integral Operators for Generalized Grand Morrey-Martingale Spaces

Hao

Ding

2022

Journal of Function Spaces

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Martingales and function spaces

Cited by 5 publications

References 46 publications

Dual Spaces for Weak Martingale Hardy Spaces Associated with Rearrangement-Invariant Spaces

Dual Spaces for Weak Martingale Hardy Spaces Associated with Rearrangement-Invariant Spaces

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

The Boundedness of Doob’s Maximal and Fractional Integral Operators for Generalized Grand Morrey-Martingale Spaces

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