The return times property for the tail on logarithm-type spaces

Carro, María Jesús Casals; Domingo-Salazar, Carlos

doi:10.3934/dcds.2018084

Cited by 2 publications

(2 citation statements)

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“…In that work, intriguing applications to ergodic theory were derived employing an extension of the restricted extrapolation theorem within the linear setting, applied to weights belonging to one-sided Sawyer's classes. In fact using similar ideas as in [9] (see also [1] and [7]) we could deduce endpoint estimates for a discrete variant of N . However, for the sake of brevity and since we can already show the potential of Theorem 4.1 with the operator N , we remain in that particular continuous context.…”

Section: Applicationsmentioning

confidence: 91%

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: Applicationsmentioning

confidence: 91%

Weighted mixed weak-type inequalities for multilinear operators

Ombrosi

Picardi

2019

Studia Math.

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“…. The current best result on the space for which the return property for the tail of the averages holds for the function 𝑓 while 𝑔 ∈ 𝐿 1 (𝜌) has been obtained by Maria Carro and C. Domingo-Salazar [9]. See also S. Baena-Miret and M. Carro*.…”

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confidence: 97%

Wiener-Wintner Ergodic Theorem, in Brief

Assani¹

2022

Notices Amer. Math. Soc.

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In 1941, N. Wiener and A. Wintner introduced a strengthening of the Birkhoff-Khinchin Ergodic Theorem, which initiated the study of a general phenomenon in ergodic theory where samplings are "good" for an uncountable number of systems. While this result was interesting on its own, it took mathematicians a few decades to realize that these "good" samplings play key roles in various other types of ergodic averages.In this short note, we will briefly introduce the theorem and discuss how it played a key role in the study of Furstenberg averages, averages along the cubes, and the Return Times Theorem.

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The return times property for the tail on logarithm-type spaces

Cited by 2 publications

References 21 publications

Weighted mixed weak-type inequalities for multilinear operators

Weighted mixed weak-type inequalities for multilinear operators

Wiener-Wintner Ergodic Theorem, in Brief

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