Weighted variation inequalities for singular integrals and commutators

Wen, Yongming; Wu, Huoxiong; Zhang, Jing

doi:10.1016/j.jmaa.2019.123825

Cited by 6 publications

(5 citation statements)

References 60 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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“…Particularly, the classical work of -variation operators for singular integrals was given in [7], in which the authors obtained the -bounds and weak type (1,1) bounds for -variation operators of truncated Hilbert transform if , and then extended to higher dimensional cases in [8]. For further studies, we refer readers to [9, 14, 17, 33, 34, 42], etc., for variation operators of singular integrals with rough kernels and weighted cases, [3, 10, 22, 31, 41, 42], etc., for variation operators of commutators.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Variational characterizations of weighted Hardy spaces and weighted spaces

Guo¹,

Wen²,

Wu³

et al. 2021

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Particularly, the classical work of ρ-variation operators for singular integrals was given in [5], in which the authors obtained the L p -bounds and weak type (1,1) bounds for ρ-variation operators of truncated Hilbert transform if ρ > 2, and then extended to higher dimensional cases in [6]. For further studies, we refer readers to [7,11,13,25,26,33] etc. for variation operators of singular integrals with rough kernels and weighted cases, [1,8,16,23,32,33] etc.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…For further studies, we refer readers to [7,11,13,25,26,33] etc. for variation operators of singular integrals with rough kernels and weighted cases, [1,8,16,23,32,33] etc. for variation operators of commutators.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Variational characterizations of weighted Hardy spaces and weighted $BMO$ spaces

Guo,

Wen,

et al. 2020

Preprint

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Weighted variation inequalities for singular integrals and commutators

Cited by 6 publications

References 60 publications

Sawyer-type inequalities for Lorentz spaces

Sawyer-type inequalities for Lorentz spaces

Variational characterizations of weighted Hardy spaces and weighted spaces

Variational characterizations of weighted Hardy spaces and weighted $BMO$ spaces

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