1996
DOI: 10.1006/jmaa.1996.0246
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Vector-Valued Extensions of Operators on Martingales

Abstract: We study weighted inequalities for vector valued extensions of the conditioned square function operator and of the maximal operators of matrix type in the case of regular martingales. As applications we obtain weighted inequalities for vectorvalued extensions of the Hardy᎐Littlewood maximal operator and of singular integral transforms on martingales.

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Cited by 6 publications
(6 citation statements)
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“…Proposition 3.6 is an extension of [24,Theorem 4.7] to the weighted setting. The result remains true for UMD Banach function spaces X and can be proved using the same techniques of [24] where one needs to apply the weighted extension of [29,Theorem 3] which is obtained in [32]. The endpoint case s = 1 of Proposition 3.6 plays a crucial rôle in the proof of Theorems 1.1 and 3.10.…”
Section: It Follows From Proposition A1 That In This Waymentioning
confidence: 87%
See 1 more Smart Citation
“…Proposition 3.6 is an extension of [24,Theorem 4.7] to the weighted setting. The result remains true for UMD Banach function spaces X and can be proved using the same techniques of [24] where one needs to apply the weighted extension of [29,Theorem 3] which is obtained in [32]. The endpoint case s = 1 of Proposition 3.6 plays a crucial rôle in the proof of Theorems 1.1 and 3.10.…”
Section: It Follows From Proposition A1 That In This Waymentioning
confidence: 87%
“…Using deep connections between harmonic analysis with weights and martingale theory, Theorem 2.6 was obtained in [2] and [29,Theorem 3] for UMD Banach function spaces in the case w = 1. It has been extended to the weighted setting in [32]. As our main result Theorem 3.10 is formulated for iterated L q (Ω)-spaces we prefer the above more elementary treatment.…”
Section: 2mentioning
confidence: 99%
“…Proof. This can be easily derived from [29,Theorem 2.6], which is a weighted version of the special case of the L p -boundedness of the Banach lattice version of the Hardy-Littlewood maximal function [8,30,58,64] for mixed-norm spaces (also see [29, Remark 2.7]).…”
mentioning
confidence: 99%
“…Proposition 3.6 is an extension of [24,Theorem 4.7] to the weighted setting. The result remains true for UMD Banach function spaces X and can be proved using the same techniques of [24] where one needs to apply the weighted extension of [29, Theorem 3] which is obtained in [32].…”
Section: Convolution Operatorsmentioning
confidence: 89%