A different approach to endpoint weak-type estimates for Calderón-Zygmund operators

Stockdale, Cody B.

doi:10.1016/j.jmaa.2020.124016

Cited by 7 publications

(7 citation statements)

References 9 publications

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“…In Theorem 7 we will generalize such a result for M, obtaining as a particular case, a new characterization of A R p and an alternative proof of the result in [15]. In [29,35], one can find similar endpoint estimates for Calderón-Zygmund operators, with p = 1 and w ∈ A 1 (see also [11,50,51,55,56]).…”

Section: Introductionmentioning

confidence: 69%

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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Section: Introductionmentioning

confidence: 69%

Sawyer-type inequalities for Lorentz spaces

Pérez

Roure-Perdices

2021

Math. Ann.

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“…This proof is motivated by the argument given by Nazarov, Treil, and Volberg in [8]. See also [10][11][12] for other applications of this technique.…”

Section: Nazarov Treil Volberg Methodsmentioning

confidence: 99%

“…The second proof is motivated by Nazarov, Treil, and Volberg's proof for the weak-type (1, 1) inequality in the nonhomogeneous setting, given in [8]. See [10][11][12] for applications of the Nazarov, Treil, and Volberg technique to multilinear and weighted settings. Refer to [5,7,9] for related results regarding multilinear and weighted Calderón-Zygmund theory.…”

Section: Introductionmentioning

confidence: 99%

A limited-range Calderón-Zygmund theorem

Grafakos,

Stockdale

2019

Preprint

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We work with singular integral operators whose kernels satisfy a condition weaker than the typical Hörmander smoothness estimate. We give two proofs of a weaktype (q, q) inequality for these operators and, via interpolation, obtain L p (R n ) estimates for the operators for a certain range of p. One proof of the weak-type estimate uses the Calderón-Zygmund decomposition while the other proof uses ideas first given by Nazarov, Treil, and Volberg.

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“…Our proof of Theorem 1.1 is an adaptation of the argument given by F. Nazarov, S. Treil, and A. Volberg in [10] and studied further in [6,[16][17][18]. While a direct application of these arguments yields exponential growth in the dimension, we here make suitable dimensional modifications and a careful accounting to remove this dependence.…”

Section: Introductionmentioning

confidence: 99%

On the dimensional weak-type $(1,1)$ bound for Riesz transforms

Spector¹,

Stockdale²

2020

Preprint

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Let R j denote the j th Riesz transform on R n . We prove that there exists an absolute constant C > 0 such thatwhere the above supremum is taken over measures of the form ν. This shows that to establish dimensional estimates for the weak-type (1, 1) inequality for the Riesz tranforms it suffices to study the corresponding weak-type inequality for Riesz transforms applied to a finite linear combination of Dirac masses. We use this fact to give a new proof of the best known dimensional upper bound, while our reduction result also applies to a more general class of Calderón-Zygmund operators.

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A different approach to endpoint weak-type estimates for Calderón-Zygmund operators

Abstract: We present a new proof of the classical weak-type (1, 1) estimate for Calderón-Zygmund operators. This proof is inspired by ideas of Nazarov, Treil, and Volberg that address the non-doubling setting. An application to a weighted weak-type inequality is also given.

Cited by 7 publications

References 9 publications

Sawyer-type inequalities for Lorentz spaces

Sawyer-type inequalities for Lorentz spaces

A limited-range Calderón-Zygmund theorem

On the dimensional weak-type $(1,1)$ bound for Riesz transforms

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