Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces

Dragičević, Oliver; Grafakos, Loukas; Pereyra, María; Petermichl, Stefanie

doi:10.5565/publmat_49105_03

Cited by 100 publications

(110 citation statements)

References 20 publications

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“…By the sharp form of Rubio de Francia's extrapolation theorem due to Dragičević, Grafakos, Pereyra and Petermichl [6], this implies the corresponding weighted L p bound,…”

Section: Introductionmentioning

confidence: 99%

The sharp weighted bound for general Calderón--Zygmund operators

2012

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For a general Calderón-Zygmund operator T on R N , it is shown thatfor all Muckenhoupt weights w ∈ A2. This optimal estimate was known as the A2 conjecture. A recent result of Pérez-Treil-Volberg reduced the problem to a testing condition on indicator functions, which is verified in this paper. The proof consists of the following elements: (i) a variant of the NazarovTreil-Volberg method of random dyadic systems with just one random system and completely without "bad" parts; (ii) a resulting representation of a general Calderón-Zygmund operator as an average of "dyadic shifts;" and (iii) improvements of the Lacey-Petermichl-Reguera estimates for these dyadic shifts, which allow summing up the series in the obtained representation.

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“…By the sharp form of Rubio de Francia's extrapolation theorem due to Dragičević, Grafakos, Pereyra and Petermichl [6], this implies the corresponding weighted L p bound,…”

Section: Introductionmentioning

confidence: 99%

The sharp weighted bound for general Calderón--Zygmund operators

2012

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“…Consult also the works of Gundy and Wheeden [9], Dragičević et al [8] and Lerner [12,13] for related results. See also the book by Wilson [25] for an overview of weighted square function inequalities.…”

Section: Introductionmentioning

confidence: 99%

Weighted square function inequalities

Oşekowski¹

2018

Publicacions Matemàtiques

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“…For 1 ≤ p < 2 the operator norm is of the order [w] 1/(p−1) Ap and this is optimal [DGPPet,Sec 4.1]. However for p > 2 the linear results obtained by extrapolation in [DGPPet] are not optimal.…”

Section: Discussionmentioning

confidence: 99%

Haar multipliers meet Bellman functions

Pereyra¹

2009

Rev. Mat. Iberoam.

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Using Bellman function techniques, we obtain the optimal dependence of the operator norms in L 2 (R) of the Haar multipliers T t w on the correspondingcharacteristic of the weight w, for t = 1, ±1/2. These results can be viewed as particular cases of estimates on homogeneous spaces L 2 (vdσ), for σ a doubling positive measure and v ∈ A d 2 (dσ), of the weighted dyadic square function S d σ . We show that the operator norms of such square functions in L 2 (vdσ) are bounded by a linear function of the A d 2 (dσ) characteristic of the weight v, where the constant depends only on the doubling constant of the measure σ. We also show an inverse estimate for S d σ . Both results are known when dσ = dx. We deduce both estimates from an estimate for the Haar multiplier (2 . The estimate for the Haar multiplier adapted to the σ measure, (T σ v ) 1/2 , is proved using Bellman functions. These estimates are sharp in the sense that the rates cannot be improved and be expected to hold for all σ, since the particular case dσ = dx, v = w, correspond to the estimates for the Haar multipliers T 1/2 w proven to be sharp.

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Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces

Cited by 100 publications

References 20 publications

The sharp weighted bound for general Calderón--Zygmund operators

The sharp weighted bound for general Calderón--Zygmund operators

Weighted square function inequalities

Haar multipliers meet Bellman functions

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