Multilinear Marcinkiewicz-Zygmund Inequalities

Carando, Daniel; Mazzitelli, Martín; Ombrosi, Sheldy

doi:10.1007/s00041-017-9563-5

Cited by 9 publications

(7 citation statements)

References 30 publications

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“…In the particular case of paraproducts, Marcinkiewicz-Zygmund inequalities were obtained by C. Benea and C. Muscalu in [1] and [2]. The results in [4] extend the previous ones in [10] and [3].…”

Section: A Vector-valued Extension Of Theorem 110supporting

confidence: 63%

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Weighted mixed weak-type inequalities for multilinear operators

Ombrosi

Picardi

2019

Studia Math.

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In this paper we present a theorem that generalizes Sawyer's classic result about mixed weighted inequalities to the multilinear context. Let w = (w1, ..., wm) and ν = w 1 m 1 ...w 1 m m , the main result of the paper sentences that under different conditions on the weights we can obtainwhere T is a multilinear Calderón-Zygmund operator. To obtain this result we first prove it for the m-fold product of the Hardy-Littlewood maximal operator M , and also for M( f )(x): the multi(sub)linear maximal function introduced in [13].As an application we also prove a vector-valued extension to the mixed weighted weak-type inequalities of multilinear Calderón-Zygmund operators.where T is Hardy-Littlewood maximal function or any Calderón-Zygmund operator.Quantitative estimates of these mixed weighted results can be found in [16]. Moreover, it was conjectured in [7] that the conclusion of the previous theorem still holds if a weaker and more general hypothesis is satisfied. That is, if we have the following conditions on the weights, u ∈ A 1 and v ∈ A ∞ . Recently, in [14] the first two authors and C. Pérez solved such conjecture. Namely, the following theorem was proved, which constitutes the more difficult case of this class of mixed weighted inequalities.

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Section: A Vector-valued Extension Of Theorem 110supporting

confidence: 63%

“…Recently in [4] D. Carando, M. Mazzitelli and the second author obtained a generalization of the Marcinkiewicz-Zygmund inequalities to the context of multilinear operators. In the particular case of paraproducts, Marcinkiewicz-Zygmund inequalities were obtained by C. Benea and C. Muscalu in [1] and [2].…”

Section: A Vector-valued Extension Of Theorem 110mentioning

confidence: 99%

Weighted mixed weak-type inequalities for multilinear operators

Ombrosi

Picardi

2019

Studia Math.

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“…Weighted norm inequalities for these operators have been considered by several authors: we refer the reader to [20,27] for precise definitions of these operators and weighted norm inequalities for them. Very recently, Carando, Mazzitelli and Ombrosi [6] proved the following weighted Marcinkiewicz-Zygmund inequalities.…”

Section: And the Conclusion Is Thatmentioning

confidence: 99%

“…Remark 1.43. In [6] the authors actually prove that Theorem 1.39 holds for weights in the larger class A p introduced in [27]. However, it is not known whether multilinear extrapolation holds for these weights.…”

Section: And the Conclusion Is Thatmentioning

confidence: 99%

Limited range multilinear extrapolation with applications to the bilinear Hilbert transform

Cruz-Uribe

Martell

2018

Math. Ann.

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We prove a limited range, off-diagonal extrapolation theorem that generalizes a number of results in the theory of Rubio de Francia extrapolation, and use this to prove a limited range, multilinear extrapolation theorem. We give two applications of this result to the bilinear Hilbert transform. First, we give sufficient conditions on a pair of weights w 1 , w 2 for the bilinear Hilbert transform to satisfy weighted norm inequalities of the formThis improves the recent results of Culiuc et al. by increasing the families of weights for which this inequality holds and by pushing the lower bound on p from 1 down to 2 3 , the critical index from the unweighted theory of the bilinear Hilbert transform. Second, as an easy consequence of our method we obtain that the bilinear Hilbert transform satisfies some vector-valued inequalities with Muckenhoupt weights. This reproves and generalizes some of the vector-valued estimates obtained by Benea and Muscalu in the unweighted case. We also generalize recent results of Carando, et al. on Marcinkiewicz-Zygmund estimates for multilinear Calderón-Zygmund operators.

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“…With this in hand, since 0 < r < max{r 1 , r 2 } < s ≤ 2 we can apply the bilinear Marcinkiewicz-Zygmund inequality in [6,Proposition 5.3] to obtain that ∞ i,j=1…”

Section: So-called T 1 Theorems Give Conditions Under Which Singular mentioning

confidence: 99%

End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications

Martell

Martikainen

et al. 2020

Trans. Amer. Math. Soc.

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In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here we consider the situations where some of the exponents of the Lebesgue spaces appearing in the hypotheses and/or in the conclusion can be possibly infinity. The scheme we follow is similar, but, in doing that, we need to develop a one-variable end-point off-diagonal extrapolation result. This complements the corresponding "finite" case obtained by Duoandikoetxea, which was one of the main tools in the aforementioned paper. The second goal of this paper is to present some applications. For example, we obtain the full range of mixed-norm estimates for tensor products of bilinear Calderón-Zygmund operators with a proof based on extrapolation and on some estimates with weights in some mixed-norm class. The same occurs with the multilinear Calderón-Zygmund operators, the bilinear Hilbert transform, and the corresponding commutators with BMO functions. Extrapolation along with the already established weighted norm inequalities easily give scalar and vector-valued inequalities with multilinear weights and these include the end-point cases.

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Multilinear Marcinkiewicz-Zygmund Inequalities

Cited by 9 publications

References 30 publications

Weighted mixed weak-type inequalities for multilinear operators

Weighted mixed weak-type inequalities for multilinear operators

Limited range multilinear extrapolation with applications to the bilinear Hilbert transform

End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications

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