Mixed Weak Estimates of Sawyer Type for Commutators of Generalized Singular Integrals and Related Operators

Berra, Fabio; Carena, Marilina; Pradolini, Gladis

doi:10.1307/mmj/1559894545

Cited by 22 publications

(27 citation statements)

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“…By a dyadic grid D we will understand a collection of cubes of R n that satisfies the following properties (1)…”

Section: Preliminaries and Basic Resultsmentioning

confidence: 99%

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Mixed weak estimates of Sawyer type for generalized maximal operators

Berra

2019

Proc. Amer. Math. Soc.

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We study mixed weak estimates of Sawyer type for maximal operators associated to the family of Young functions Φ(t) = t r (1 + log + t) δ , where r ≥ 1 and δ ≥ 0. More precisely, if u and v r are A1 weights, and w is defined as w = 1/Φ(v −1 ) then the following estimateholds for every positive t. This extends mixed estimates to a wider class of maximal operators, since when we put r = 1 and δ = 0 we recover a previous result for the classical Hardy-Littlewood maximal operator. This inequality generalizes the result proved by Sawyer in Proc. Amer. Math. Soc. 93 (1985), no. 4, 610-614. Moreover, it includes estimates for some maximal operators related with commutators of Calderón-Zygmund operators.2010 Mathematics Subject Classification. 42B20, 42B25.

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“…By a dyadic grid D we will understand a collection of cubes of R n that satisfies the following properties (1)…”

Section: Preliminaries and Basic Resultsmentioning

confidence: 99%

“…In [1] the authors proved a mixed weighted inequality for such operators, but for a particular weight v(x) = |x| −β with β < −n. This means that v is not even locally integrable.…”

Section: Introductionmentioning

confidence: 99%

Mixed weak estimates of Sawyer type for generalized maximal operators

Berra

2019

Proc. Amer. Math. Soc.

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“…For example, it is well‐known that certain maximal operators associated to the Young function

φ false(t false) = t {(1 + \log^{+} t)}^{m}

control the higher order commutators of CZOs. In this direction, in [5], the authors proved mixed weak estimates in

R^{n}

for weights u and v , where u is arbitrary but

v = {| x |}^{β}

with

β < - n

. Concretely, we proved that

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}) M u false(x false) dx

holds for every positive t , where

w = 1 / {normalΦ}_{0} (v^{- 1})

{normalΦ}_{0} false(t false) = t^{r} {(1 + \log^{+} t)}^{δ}

, with

r \geq 1

and

δ \geq 0

.…”

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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“…for a norm or a quasi-norm • X , and v −1 ∈ A p . Further results for commutators involving Sawyer-type inequalities can be found in [6,7] (see also [8,9]).…”

Section: Introductionmentioning

confidence: 89%

Sawyer-type inequalities for Lorentz spaces

Pérez

Roure-Perdices

2021

Math. Ann.

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The Hardy-Littlewood maximal operator M satisfies the classical Sawyer-type estimate $$\begin{aligned} \left\| \frac{Mf}{v}\right\| _{L^{1,\infty }(uv)} \le C_{u,v} \Vert f \Vert _{L^{1}(u)}, \end{aligned}$$ Mf v L 1 , ∞ ( u v ) ≤ C u , v ‖ f ‖ L 1 ( u ) , where $$u\in A_1$$ u ∈ A 1 and $$uv\in A_{\infty }$$ u v ∈ A ∞ . We prove a novel extension of this result to the general restricted weak type case. That is, for $$p>1$$ p > 1 , $$u\in A_p^{{\mathcal {R}}}$$ u ∈ A p R , and $$uv^p \in A_\infty $$ u v p ∈ A ∞ , $$\begin{aligned} \left\| \frac{Mf}{v}\right\| _{L^{p,\infty }(uv^p)} \le C_{u,v} \Vert f \Vert _{L^{p,1}(u)}. \end{aligned}$$ Mf v L p , ∞ ( u v p ) ≤ C u , v ‖ f ‖ L p , 1 ( u ) . From these estimates, we deduce new weighted restricted weak type bounds and Sawyer-type inequalities for the m-fold product of Hardy-Littlewood maximal operators. We also present an innovative technique that allows us to transfer such estimates to a large class of multi-variable operators, including m-linear Calderón-Zygmund operators, avoiding the $$A_\infty $$ A ∞ extrapolation theorem and producing many estimates that have not appeared in the literature before. In particular, we obtain a new characterization of $$A_p^{{\mathcal {R}}}$$ A p R . Furthermore, we introduce the class of weights that characterizes the restricted weak type bounds for the multi(sub)linear maximal operator $${\mathcal {M}}$$ M , denoted by $$A_{\mathbf {P}}^{{\mathcal {R}}}$$ A P R , establish analogous bounds for sparse operators and m-linear Calderón-Zygmund operators, and study the corresponding multi-variable Sawyer-type inequalities for such operators and weights. Our results combine mixed restricted weak type norm inequalities, $$A_p^{{\mathcal {R}}}$$ A p R and $$A_{\mathbf {P}}^{{\mathcal {R}}}$$ A P R weights, and Lorentz spaces.

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Mixed Weak Estimates of Sawyer Type for Commutators of Generalized Singular Integrals and Related Operators

Cited by 22 publications

References 20 publications

Mixed weak estimates of Sawyer type for generalized maximal operators

Mixed weak estimates of Sawyer type for generalized maximal operators

Improvements on Sawyer type estimates for generalized maximal functions

Sawyer-type inequalities for Lorentz spaces

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