A characterization of the weighted weak type Coifman–Fefferman and Fefferman–Stein inequalities

Lerner, Andrei K.

doi:10.1007/s00208-020-01980-z

Cited by 9 publications

(4 citation statements)

References 31 publications

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“…In the last years some advances have been made in the study of this kind of questions. Lerner [16] fully characterized the weak type version of (1.1). Sawyer's result has been extended to the full range in [6] and also quantitative estimates in terms of a suitable constant and further operators, such as rough singular integrals, have been explored in [4, 5].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

One-sided estimates via function

Lorente¹,

Martín-Reyes²,

Rivera-Rı́os³

2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We recall that $w\in C_{p}^+$ if there exist $\varepsilon >0$ and $C>0$ such that for any $a< b< c$ with $c-b< b-a$ and any measurable set $E\subset (a,b)$ , the following holds \[ \int_{E}w\leq C\left(\frac{|E|}{(c-b)}\right)^{\varepsilon}\int_{\mathbb{R}}\left(M^+\chi_{(a,c)}\right)^{p}w<\infty. \] This condition was introduced by Riveros and de la Torre [33] as a one-sided counterpart of the $C_{p}$ condition studied first by Muckenhoupt and Sawyer [30, 34]. In this paper we show that given $1< p< q<\infty$ if $w\in C_{q}^+$ then \[ \|M^+f\|_{L^{p}(w)}\lesssim\|M^{\sharp,+}f\|_{L^{p}(w)} \] and conversely if such an inequality holds, then $w\in C_{p}^+$ . This result is the one-sided counterpart of Yabuta's main result in [37]. Combining this estimate with known pointwise estimates for $M^{\sharp,+}$ in the literature we recover and extend the result for maximal one-sided singular integrals due to Riveros and de la Torre [33] obtaining counterparts a number of operators.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

One-sided estimates via function

Lorente¹,

Martín-Reyes²,

Rivera-Rı́os³

2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…In the last years several advances have been made, for instance the extension in [5] to the full range 0 < p < ∞ and other operators relying upon [45,22] and sparse domination techniques, the characterization of the good weights for the weak type counterpart of (2.5) in [25], and the quantitative results introduced in [3] and further explored in [4].…”

Section: 1mentioning

confidence: 99%

Quantitative matrix weighted estimates for certain singular integral operators

Muller¹,

Rivera-Rı́os²

2021

Preprint

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In this paper quantitative weighted matrix estimates for vector valued extensions of L r ′ -Hörmander operators and rough singular integrals are studied. Strong type (p, p) estimates, endpoint estimates, and some new results on Coifman-Fefferman estimates assuming A∞ and Cp condition counterparts are provided. To prove the aforementioned estimates we rely upon some suitable convex body domination results that we settle as well in this paper.

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“…Although C p weights were introduced in the context of the CFI, other inequalities have been proved to hold for these weights. For example, the Fefferman-Stein inequality, between the maximal operators of Hardy-Littlewood and of Fefferman-Stein, as can be found in [27], [6] for a quantified version, [17] in the weak-type context. In [7], the authors extended Sawyer's result to a wider class of operators than Calderón-Zygmund operators, including some pseudo-differential operators and oscillatory integrals.…”

Section: P Weights and The Coifman-fefferman Inequalitymentioning

confidence: 99%

Sharp Reverse Hölder Inequality for $$C_p$$ Weights and Applications

Canto

2020

J Geom Anal

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We prove an appropriate sharp quantitative reverse Hölder inequality for the C p class of weights from which we obtain as a limiting case the sharp reverse Hölder inequality for the A ∞ class of weights [13, 14]. We use this result to provide a quantitative weighted norm inequality between Calderón-Zygmund operators and the Hardy-Littlewood maximal function, precisely T f L p (w) T,n,p,q [w] Cq (1 + log + [w] Cq) M f L p (w) , for w ∈ C q and q > p > 1, quantifying Sawyer's theorem [26].

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A characterization of the weighted weak type Coifman–Fefferman and Fefferman–Stein inequalities

Abstract: A condition on a Banach function space X is given under which the Coifman-Fefferman and Fefferman-Stein inequalities on X are equivalent.2010 Mathematics Subject Classification. 42B20, 42B25.

Cited by 9 publications

References 31 publications

One-sided estimates via function

One-sided estimates via function

Quantitative matrix weighted estimates for certain singular integral operators

Sharp Reverse Hölder Inequality for $$C_p$$ Weights and Applications

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