On the A 2 Inequality for Calderón–Zygmund Operators

Lacey, Michael T.

doi:10.1007/978-1-4614-4565-4_20

Cited by 5 publications

(4 citation statements)

References 6 publications

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“…After that the proof reduces to showing (2.1) for S m,k D in place of T with the corresponding constant depending linearly (or polynomially) on the complexity. Observe that over the past year several different proofs of the latter step appeared (see, e.g., [15,26]).…”

Section: 1mentioning

confidence: 99%

On an estimate of Calderón-Zygmund operators by dyadic positive operators

Lerner

2013

JAMA

134

136

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Given a general dyadic grid D and a sparse family ofGiven a Banach function space X(R n ) and the maximal Calderón-Zygmund operator T ♮ , we show thatThis result is applied to weighted inequalities. In particular, it implies: (i) the "two-weight conjecture" by D. Cruz-Uribe and C. Pérez in full generality; (ii) a simplification of the proof of the "A 2 conjecture"; (iii) an extension of certain mixed A p -A r estimates to general Calderón-Zygmund operators; (iv) an extension of sharp A 1 estimates (known for T ) to the maximal Calderón-Zygmund operator T ♮ .

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Section: 1mentioning

confidence: 99%

On an estimate of Calderón-Zygmund operators by dyadic positive operators

Lerner

2013

JAMA

134

136

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“…The second key element of all known proofs was showing (1.1) for S m,k D in place of T with the corresponding constant depending linearly (or polynomially) on the complexity. Observe that over the past year several different proofs of this step appeared (see, e.g., [15,25]).…”

Section: Introductionmentioning

confidence: 99%

A Simple Proof of the A2 Conjecture

Lerner

2012

International Mathematics Research Notices

237

207

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“…Since the appearance of Hytönen's theorem several simplifications of the argument have appeared [39,59,43,75,35], as well as an extension to metric spaces with geometric doubling condition [60]. There is also a very nice survey of the A 2 conjecture [42].…”

Section: The a 2 Conjecture (Now Theorem)mentioning

confidence: 99%

“…Precursors to Petermichl's and Hytönen's results can be found in Figiel's work [25]. Nowadays some of the simpler arguments yielding polynomial and even linear dependence on the complexity use minimally Bellman functions [59,75], or do not use them at all [37,43]. The commutator [b, H] is more singular than the operator H and this is reflected on the nature of its bounds on weighted L p -spaces.…”

Section: Introductionmentioning

confidence: 99%

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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On the A 2 Inequality for Calderón–Zygmund Operators

Abstract: Abstract. We prove that for an L 2 (R d )-bounded Calderón-Zygmund operator and weight w ∈ A 2 , that we have the inequality below due to Hytönen,Our proof will appeal to a distributional inequality used by several authors, adapted Haar functions, and standard stopping times.

Cited by 5 publications

References 6 publications

On an estimate of Calderón-Zygmund operators by dyadic positive operators

On an estimate of Calderón-Zygmund operators by dyadic positive operators

A Simple Proof of the A2 Conjecture

Weighted Inequalities and Dyadic Harmonic Analysis

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