An estimate for weighted Hilbert transform via square functions

Petermichl, Stefanie; Pott, Sandra

doi:10.1090/s0002-9947-01-02938-5

Cited by 25 publications

(23 citation statements)

References 6 publications

(9 reference statements)

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“…This problem remained open for a few years and was finally answered by Hukovič, Treil, and Volberg [10]: the exponent 1 is the best. See also the paper [20] by Petermichl and Pott, and [26] by Wittwer. This result was significantly extended by Cruz-Uribe, Martell, and Pérez [5] to the case of general A p weights.…”

Section: Introductionmentioning

confidence: 99%

Weighted square function inequalities

Oşekowski¹

2018

Publicacions Matemàtiques

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Section: Introductionmentioning

confidence: 99%

Weighted square function inequalities

Oşekowski¹

2018

Publicacions Matemàtiques

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“…It turns out that the sharp dependence is linear, i.e., the best exponent is 1. This result was later reproved by Wittwer [49] and Petermichl and Pott in [37] using a different approach. Actually, the latter paper contains also the proof of the reverse inequality…”

Section: Introductionmentioning

confidence: 92%

Weighted 𝐿² inequalities for square functions

Bañuelos

Oşekowski

2017

Trans. Amer. Math. Soc.

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Using Bellman function approach, we present new proofs of weighted L 2 inequalities for square functions, with the optimal dependence on the A 2 characteristics of the weight and further explicit constants. We study the estimates both in the analytic and probabilistic context, and, as application, obtain related estimates for the classical Lusin and Littlewood-Paley square functions.where the summation runs over all nonnegative integers n such that x ∈ I n .

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“…and the best power of [w] A p is at least max 1, 1 p−1 . In the case p = 2 and T = H is the Hilbert transform, Petermichl and Pott improved it to the power 3 2 (see [28]) and, later on, Petermichl in [25] obtained the best possible linear dependence for p ≥ 2. For the case 1 < p < 2, that dependence of the [w] A p constant is a consequence of the sharp extrapolation proved in [5].…”

Section: And We Define [W]mentioning

confidence: 99%

Mixed estimates for singular integrals on weighted Hardy spaces

Cejas

Dalmasso

2019

Rev Mat Complut

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In this paper we give quantitative bounds for the norms of different kinds of singular integral operators on weighted Hardy spaces H p w , where 0 < p ≤ 1 and w is a weight in the Muckenhoupt A ∞ class. We deal with Fourier multiplier operators, Calderón-Zygmund operators of homogeneous type which are particular cases of the first ones, and, more generally, we study singular integrals of convolution type. In order to prove mixed estimates in the setting of weighted Hardy spaces, we need to introduce several characterizations of weighted Hardy spaces by means of square functions, Littlewood-Paley functions and the grand maximal function. We also establish explicit quantitative bounds depending on the weight w when switching between the H p w -norms defined by the Littlewood-Paley-Stein square function and its discrete version, and also by applying the mixed bound A q −A ∞ result for the vector-valued extension of the Hardy-Littlewood maximal operator given in Buckley (Trans Am Math Soc 340(1): [253][254][255][256][257][258][259][260][261][262][263][264][265][266][267][268][269][270][271][272] 1993).

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An estimate for weighted Hilbert transform via square functions

Cited by 25 publications

References 6 publications

Weighted square function inequalities

Weighted square function inequalities

Weighted 𝐿² inequalities for square functions

Mixed estimates for singular integrals on weighted Hardy spaces

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