Mixed weak estimates of Sawyer type for generalized maximal operators

Berra, Fabio

doi:10.1090/proc/14495

Cited by 16 publications

(20 citation statements)

References 9 publications

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“…That conjecture was positively answered recently in [24] where several quantitative estimates were provided as well. At this point we would like to mention, as well, a recent generalization provided for Orlicz maximal operators in [2].In [7], besides the aforementioned results, it was shown that (1.1) holds if u ∈ A 1 and v ∈ A ∞ (u) (see Section 2.2 for the precise definition of A p (u)). The advantage of that condition is that the product uv is an A ∞ weight.…”

mentioning

confidence: 88%

A sparse approach to mixed weak type inequalities

Caldarelli

Rivera-Rı́os

2019

Math. Z.

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In this paper we provide some quantitative mixed-type estimates assuming conditions that imply that uv ∈ A∞ for Calderón-Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced in [8] and extended in [23] and [25]Sawyer also conjectured that (1.1) should hold as well for the Hilbert transform. Cruz-Uribe, Martell and Pérez [7] generalized (1.1) to higher dimensions and actually proved that Sawyer's conjecture holds for Calderón-Zygmund operators via the following extrapolation argument.Theorem C. Assume that for every w ∈ A ∞ and some 0 < p < ∞,Then for every u ∈ A 1 and everyThe conditions on the weights in that extrapolation result lead them to conjecture that (1.1), and consequently the corresponding estimate for Calderón-Zygmund operators should hold as well with u ∈ A 1 and v ∈ A ∞ . That conjecture was positively answered recently in [24] where several quantitative estimates were provided as well. At this point we would like to mention, as well, a recent generalization provided for Orlicz maximal operators in [2].In [7], besides the aforementioned results, it was shown that (1.1) holds if u ∈ A 1 and v ∈ A ∞ (u) (see Section 2.2 for the precise definition of A p (u)). The advantage of that condition is that the product uv is an A ∞ weight. Over the past few years, there have been new contributions under those assumptions such as [3] for the case of fractional integrals and related operators, [28,29] for related quantitative estimates and [26] for multilinear extensions.

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confidence: 88%

A sparse approach to mixed weak type inequalities

Caldarelli

Rivera-Rı́os

2019

Math. Z.

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“…Later on, in [4] the author showed that a similar behavior occurs in the case

u, v^{r} \in A_{1}

, that is,

u w (({} x \in {double-struckR}^{n} : \frac{M_{Φ_{0}} false(f v false) false(x false)}{v (x)} > t)) \leq C \int_{R^{n}} {normalΦ}_{0} (\frac{false| f false(x false) false| v false(x false)}{t}) u false(x false) dx,

where w and Φ 0 are as above.…”

Section: Introductionmentioning

confidence: 85%

“…A motivation for studying these type of estimates is to find an alternative way to prove the boundedness of the operator

M_{Φ}

. In [4] was established that (1.4) extends the estimates in [6] not only for M but also for

M_{r}

. However, this inequality turns out to be non‐homogeneous, and even when

v^{r} \in A_{1}

, the resulting weight w might not.…”

Section: Introductionmentioning

confidence: 99%

Improvements on Sawyer type estimates for generalized maximal functions

Berra

Carena

Pradolini

2020

Mathematische Nachrichten

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In this paper we prove mixed inequalities for the maximal operator Φ , for general Young functions Φ with certain additional properties, improving and generalizing some previous estimates for the Hardy-Littlewood maximal operator proved by E. Sawyer. We show that given ≥ 1, if , are weights belonging to the 1-Muckenhoupt class and Φ is a Young function as above, then the inequality ({ ∈ ℝ ∶ Φ ()() () > }) ≤ ∫ ℝ Φ (| ()|) () () dx holds for every positive. A motivation for studying these type of estimates is to find an alternative way to prove the boundedness properties of Φ. Moreover, it is well-known that for the particular case Φ() = (1 + log +) with ∈ ℕ these maximal functions control, in some sense, certain operators in harmonic analysis.

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“…holds for every positive t, where w = 1/Φ 0 (v −1 ), Φ 0 (t) = t r (1 + log + t) δ , with r ≥ 1 and δ ≥ 0. Later on, in [3] the author showed that a similar behavior occurs in the case u, v r ∈ A 1 , that is,…”

Section: Introductionmentioning

confidence: 80%

Improvements on Sawyer type estimates for generalized maximal functions

Berra¹,

Carena²,

Pradolini³

2019

Preprint

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In this paper we prove mixed inequalities for the maximal operator MΦ, for general Young functions Φ with certain additional properties, improving and generalizing some previous estimates for the Hardy-Littlewood maximal operator proved by E. Sawyer. We show that given r ≥ 1, if u, v r are weights belonging to the A1-Muckenhoupt class and Φ is a Young function as above, then the inequalityholds for every positive t.A motivation for studying these type of estimates is to find an alternative way to prove the boundedness properties of MΦ. Moreover, it is well-known that for the particular case Φ(t) = t(1 + log + t) m with m ∈ N these maximal functions control, in some sense, certain operatos in Harmonic Analysis.

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Mixed weak estimates of Sawyer type for generalized maximal operators

Cited by 16 publications

References 9 publications

A sparse approach to mixed weak type inequalities

A sparse approach to mixed weak type inequalities

Improvements on Sawyer type estimates for generalized maximal functions

Improvements on Sawyer type estimates for generalized maximal functions

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