Weighted weak-type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> estimates via Rubio de Francia extrapolation

Carro, María Jesús Casals; Grafakos, Loukas; Soria, Javier

doi:10.1016/j.jfa.2015.06.005

Cited by 32 publications

(13 citation statements)

References 16 publications

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“…These classes of weights are closely related to the A p class introduced by Muckenhoupt in [20]. For our purposes, a fundamental property of A R p is the following [12]. Lemma 2.1.…”

Section: Theorem 13 ([4]mentioning

confidence: 91%

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

Carro¹,

Domingo-Salazar²

2018

Discrete &Amp; Continuous Dynamical Systems - A

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“…These classes of weights are closely related to the A p class introduced by Muckenhoupt in [20]. For our purposes, a fundamental property of A R p is the following [12]. Lemma 2.1.…”

Section: Theorem 13 ([4]mentioning

confidence: 91%

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

Carro¹,

Domingo-Salazar²

2018

Discrete &Amp; Continuous Dynamical Systems - A

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“…, m, and v = ν −1/ p w . Indeed, in virtue of Theorem 2.7 in [13], we have that w i ∈ A R p i , i = 1, . .…”

Section: Remarkmentioning

confidence: 95%

“…Recently, E. R. P. has shown in [53] that Sawyer-type inequalities for Lorentz spaces play a fundamental role in the extension to the multi-variable setting of the restricted weak type Rubio de Francia's extrapolation presented in [13,15]. His approach suggests that Conjecture 1 will be crucial for proving multi-variable extrapolation theorems involving weights in A R P .…”

Section: Introductionmentioning

confidence: 99%

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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“…Concerning weighted bounds for

M^{\otimes}

, the above easy exercise becomes an open question when we want to characterize the weights for which

M^{\otimes} : L^{p_{1}, 1} false(w_{1} false) \times L^{p_{2}, 1} false(w_{2} false) ⟶ L^{p, \infty} (w_{1}^{p / p_{1}} w_{2}^{p / p_{2}}),

and this is the question we want to address in this paper. In fact, the motivation comes from the recent restricted weak type Rubio de Francia extrapolation theory (see , ) where it has been proved that if an operator

T : L^{p, 1} false(w false) ⟶ L^{p, \infty} false(w false),

for every

w \in A_{p}^{scriptR}

, then endpoint (1, 1) estimates hold for characteristic functions; that is

false∥ T χ_{E} ∥_{L^{1, \infty} false(u false)} ≲ u false(E false), for all u \in A_{1},

contrary to what happens with the classical Rubio de Francia theory. In this context, it has been proved in that the operator

M^{\otimes}

plays in the multilinear extrapolation theory of Rubio de Francia (see ) the same role that the classical Hardy–Littlewood maximal operator plays in the linear case of this theory.…”

Section: Introductionmentioning

confidence: 99%

“…, and this is the question we want to address in this paper. In fact, the motivation comes from the recent restricted weak type Rubio de Francia extrapolation theory (see [2], [3]) where it has been proved that if an operator…”

Section: Introductionmentioning

confidence: 99%

Weighted boundedness of the 2‐fold product of Hardy–Littlewood maximal operators

Carro

Roure

2018

Mathematische Nachrichten

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We study new weighted estimates for the 2‐fold product of Hardy–Littlewood maximal operators defined by M⊗false(f,gfalse):=MfMg. This operator appears very naturally in the theory of bilinear operators such as the bilinear Calderón–Zygmund operators, the bilinear Hardy–Littlewood maximal operator introduced by Calderón or in the study of pseudodifferential operators. To this end, we need to study Hölder's inequality for Lorentz spaces with change of measures trueright72.0ptfalse∥fgfalse∥Lp,∞()w1p/p1w2p/p2≤Cfalse∥ffalse∥Lp1,∞(w1)false∥gfalse∥Lp2,∞(w2).Unfortunately, we shall prove that this inequality does not hold, in general, and we shall have to consider a weaker version of it.

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Weighted weak-type (1,1) estimates via Rubio de Francia extrapolation

Cited by 32 publications

References 16 publications

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

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

Sawyer-type inequalities for Lorentz spaces

Weighted boundedness of the 2‐fold product of Hardy–Littlewood maximal operators

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