Erratum: Fractional Integral on Martingale Hardy Spaces With Variable Exponents

Jiao, Yong; Zhou, Dejian; Weisz, Ferenc; Hao, Zhiwei

doi:10.1515/fca-2017-0055

Cited by 13 publications

(6 citation statements)

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“…The fractional maximal operator M α generates the Hardy-Littlewood operator and is bounded from L q(•) to L p(•) , whenever 0 ≤ α < 1, 1 < q − ≤ q + ≤ 1/α and 1/ p(•) = 1/q(•) − α (see Capone et al [2], Cruz-Uribe and Fiorenza [5] or Kokilashvili and Meskhi [22]). The fractional integral operator was investigated in Diening [7], Ephremidze et al [9], Mizuta and Shimomura [28], Rafeiro and Samko [35], Izuki et al [15] and, for martingales, in Arai et al [1], Hao et al [13,20], Jiao et al [17] and Sadasue [36].…”

Section: Introductionmentioning

confidence: 99%

New fractional maximal operators in the theory of martingale Hardy and Lebesgue spaces with variable exponents

Weisz

2022

Fract Calc Appl Anal

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We generalize the usual Doob maximal operator as well as the fractional maximal operator and introduce $$M_{\gamma ,s,\alpha }$$ M γ , s , α , a new fractional maximal operator for martingales. We prove that under the log-Hölder continuity condition of the variable exponents $$p(\cdot )$$ p ( · ) and $$q(\cdot )$$ q ( · ) , the maximal operator $$M_{\gamma ,s,\alpha }$$ M γ , s , α is bounded from the variable Lebesgue space $$L_{q(\cdot )}$$ L q ( · ) to $$L_{p(\cdot )}$$ L p ( · ) and from the variable Hardy space $$H_{q(\cdot )}$$ H q ( · ) to $$L_{p(\cdot )}$$ L p ( · ) , whenever $$0 \le \alpha <1$$ 0 ≤ α < 1 , $$0<q_-\le q_+ \le 1/\alpha $$ 0 < q - ≤ q + ≤ 1 / α , $$0<\gamma ,s<\infty $$ 0 < γ , s < ∞ , $$1/p(\cdot )= 1/q(\cdot )- \alpha $$ 1 / p ( · ) = 1 / q ( · ) - α and $$1/p_- - 1/p_+ < \gamma +s$$ 1 / p - - 1 / p + < γ + s . Moreover, for $$\alpha =0$$ α = 0 , the operator $$M_{\gamma ,s,0}$$ M γ , s , 0 generates equivalent quasi-norms on the Hardy spaces $$H_{p(\cdot )}$$ H p ( · ) .

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Section: Introductionmentioning

confidence: 99%

New fractional maximal operators in the theory of martingale Hardy and Lebesgue spaces with variable exponents

Weisz

2022

Fract Calc Appl Anal

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“…We refer the reader to recent monographs [32,63,64] for more discussions on martingale inequalities. On another hand, various martingale Hardy spaces were considered in many articles, for instance, martingale Hardy spaces, martingale Lorentz Hardy spaces and martingale variable Hardy spaces were studied by Weisz [76,77,75] and Jiao et al [49,40,39,35,38]. Moreover, martingale Lorentz-Karamata Hardy spaces and multi-parameter martingale Hardy spaces were investigated by Ho [29], Jiao et al [36] and Weisz [75].…”

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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“…For general martingale Hardy spaces, instead of the log-Hölder continuity condition, we supposed in [9,11,13,33,36] the slightly more general condition (2.2) for all atoms of the σ-algebras. All results of this paper remain true if, instead of (2.1), we suppose (2.2) for all dyadic intervals I ⊂ [0, 1).…”

Section: F Weiszmentioning

confidence: 99%

“…Cruz-Uribe and Fiorenza [6] and Diening et al [7]). The fractional integral operator was investigated in Ephremidze et al [8], Rafeiro and Samko [27,25,26,30] and, for martingales, in Hao et al [9,13].…”

Section: Introductionmentioning

confidence: 99%

Characterizations of Variable Martingale Hardy Spaces Via Maximal Functions

Weisz

2021

Fract Calc Appl Anal

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We introduce a new type of dyadic maximal operators and prove that under the log-Hölder continuity condition of the variable exponent p(⋅), it is bounded on L p(⋅) if 1 < p − ≤ p + ≤ ∞. Moreover, the space generated by the L p(⋅)-norm (resp. the L p(⋅), q -norm) of the maximal operator is equivalent to the Hardy space H p(⋅) (resp. to the Hardy-Lorentz space H p(⋅), q ). As special cases, our maximal operator contains the usual dyadic maximal operator and four other maximal operators investigated in the literature.

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Erratum: Fractional Integral on Martingale Hardy Spaces With Variable Exponents

Abstract: This is an erratum to the paper “Fractional integral on martingale Hardy spaces with variable exponents”,

Cited by 13 publications

References 4 publications

New fractional maximal operators in the theory of martingale Hardy and Lebesgue spaces with variable exponents

New fractional maximal operators in the theory of martingale Hardy and Lebesgue spaces with variable exponents

New martingale inequalities and applications to Fourier analysis

Characterizations of Variable Martingale Hardy Spaces Via Maximal Functions

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