Lecture notes on dyadic harmonic analysis

Pereyra, María

doi:10.1090/conm/289/04874

Cited by 27 publications

(27 citation statements)

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“…Choosing λ ≥ (4C) 1/p−1 , we obtain a contradiction to (10). This proves that for λ ≥ (4C) 1/p−1 , there exists a constant 0 < c < 1 such that |G(J)| ≤ c |J| for all J ∈ D. It then follows by induction that |G k (J)| ≤ c k |J| for k ∈ N and J ∈ D.…”

Section: Weights With Reverse Hölder Property and Decaying Stopping Timementioning

confidence: 82%

“…We use a slightly different route from [2], [7]. Instead of using the weighted maximal function, we show first that the weighted square function operator is bounded and bounded below by means of a stopping time argument from [10]. This gives us the uniform boundedness of the weighted dyadic martingale transforms.…”

Section: /2mentioning

confidence: 99%

“…The proof is an adaption of the proof of the Weight Lemma 3.17 for scalar weights in [10]. We first introduce an auxiliary stopping time G. For I ∈ D, let G(I) denote the collection of all maximal dyadic subintervals J of I such that (5) holds.…”

Section: Weights With Reverse Hölder Property and Decaying Stopping Timementioning

confidence: 99%

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A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert transform

Pot

2007

Studia Math.

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Section: Weights With Reverse Hölder Property and Decaying Stopping Timementioning

confidence: 82%

Section: /2mentioning

confidence: 99%

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A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert transform

Pot

2007

Studia Math.

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“…In this note we want to highlight the interplay with dyadic harmonic analysis [62] in the solution of the A 2 conjecture. Initially the A 2 conjecture was shown to hold, one at a time, for dyadic operators and for operators such as the Hilbert transform that have lots of symmetries.…”

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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“…Apart from one main new result (a local T(b) theorem), we shall mostly take existing results (atomic decompositions, paraproduct estimates, Carleson embedding) and re-prove them in a framework which unifies both the Carleson measure theory and the theory of trees and tiles. (As such there is some overlap with the recent lecture notes in [49]). …”

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confidence: 99%

Carleson measures, trees, extrapolation, and T(b) theorems

Auscher¹,

Hofmann²,

Muscalu³

et al. 2002

Publicacions Matemàtiques

168

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Abstract. The theory of Carleson measures, stopping time arguments, and atomic decompositions has been well-established in harmonic analysis. More recent is the theory of phase space analysis from the point of view of wave packets on tiles, tree selection algorithms, and tree size estimates. The purpose of this paper is to demonstrate that the two theories are in fact closely related, by taking existing results and reproving them in a unified setting. In particular we give a dyadic version of extrapolation for Carleson measures, as well as a two-sided local dyadic T(b) theorem which generalizes earlier T(b) theorems of David, Journe, Semmes, and Christ.

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Lecture notes on dyadic harmonic analysis

Cited by 27 publications

References 0 publications

A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert transform

A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert transform

Weighted Inequalities and Dyadic Harmonic Analysis

Carleson measures, trees, extrapolation, and T(b) theorems

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