Extrapolation of weights revisited: New proofs and sharp bounds

Duoandikoetxea, Javier

doi:10.1016/j.jfa.2010.12.015

Cited by 111 publications

(114 citation statements)

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“…The value of the constant K(n, p, p 0 , [w] Ap ) first appeared in Dragičević, Grafakos, Pereyra, and Petermichl [98]. Another proof with sharp bounds (in terms of the characteristic constant of the weights) was given by Duoandikoetxea [101]. The present treatment of Theorem 7.5.3, based on crucial Lemma 7.5.4, was communicated to the author by J. M. Martell.…”

Section: Historical Notesmentioning

confidence: 93%

Classical Fourier Analysis

Grafakos¹

2014

Graduate Texts in Mathematics

1,087

534

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Section: Historical Notesmentioning

confidence: 93%

Classical Fourier Analysis

Grafakos¹

2014

Graduate Texts in Mathematics

1,087

534

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“…Duoandicoetxea found recently a shorter proof of this theorem [23]. Sharp extrapolation from r = 2 is sharp for the martingale, Hilbert, Beurling-Ahlfors and Riesz transforms for all 1 < p < ∞ [21].…”

Section: Estimates In L P (W) Via Sharp Extrapolationmentioning

confidence: 94%

Weighted Inequalities and Dyadic Harmonic Analysis

Pereyra

2012

Excursions in Harmonic Analysis, Volume 2

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We survey the recent solution of the so-called A 2 conjecture, all Calderón-Zygmund singular integral operators are bounded on L 2 (w) with a bound that depends linearly on the A 2 characteristic of the weight w, as well as corresponding results for commutators. We highlight the interplay of dyadic harmonic analysis in the solution of the A 2 conjecture, especially Hytönen's representation theorem for Calderón-Zygmund singular integral operators in terms of Haar shift operators. We describe Chung's dyadic proof of the corresponding quadratic bound on L 2 (w) for the commutator of the Hilbert transform with a BMO function, and we deduce sharpness of the bounds for the dyadic paraproduct on L p (w) that were obtained extrapolating Beznosova's linear bound on L 2 (w). We show that if an operator T is bounded on the weighted Lebesgue space L r (w) and its operator norm is bounded by a power α of the A r characteristic of the weight, then its commutator [T, b] with a function b in BMO will be bounded on L r (w) with an operator norm bounded by the increased power α + max{1, 1 r−1 } of the A r characteristic of the weight. The results are sharp in terms of the growth of the operator norm with respect to the A r characteristic of the weight for all 1 < r < ∞.

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“…Along these lines, we have to also mention the recent work [9] where a simple proof of the last result has been presented. As said in that paper, the three basic ingredients of any of the proofs of the extrapolation results are:…”

Section: Introductionmentioning

confidence: 93%

“…In fact, the main idea in [9] was to use a new factorization of A p 0 weights, while (ii) and (iii) were used in a standard way. We recall that the usual factorization result for A p 0 weights says that w ∈ A p 0 if, and only if, there exist u 0 , u 1 ∈ A 1 such that w = u 0 u 1−p 0 1 , while the argument in [9] uses the fact that, if w ∈ A p and u ∈ A 1 , then wu p−p 0 ∈ A p 0 , 1 ≤ p ≤ p 0 < ∞, and…”

Section: Introductionmentioning

confidence: 99%