Convex body domination and weighted estimates with matrix weights

Nazarov, Fëdor; Petermichl, Stefanie; Treil, Sergei; Volberg, Alexander

doi:10.1016/j.aim.2017.08.001

Cited by 48 publications

(72 citation statements)

References 18 publications

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“…Convex body domination was introduced by F. Nazarov, S. Petermichl and A. Volberg in [25]. That notion provides a suitable counterpart to sparse domination in the vector-valued setting.…”

Section: Convex Body Domination For Commutatorsmentioning

confidence: 99%

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Sharp $$A_{1}$$ Weighted Estimates for Vector-Valued Operators

Isralowitz

Pott²,

Rivera-Rı́os³

2020

J Geom Anal

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Given 1 ≤ q < p < ∞, quantitative weighted L p estimates, in terms of Aq weights, for vector valued maximal functions, Calderón-Zygmund operators, commutators and maximal rough singular integrals are obtained. The results for singular operators will rely upon suitable convex body domination results, which in the case of commutators will be provided in this work, obtaining as a byproduct a new proof for the scalar case as well.

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“…Convex body domination was introduced by F. Nazarov, S. Petermichl and A. Volberg in [25]. That notion provides a suitable counterpart to sparse domination in the vector-valued setting.…”

Section: Convex Body Domination For Commutatorsmentioning

confidence: 99%

“…Quite recently F. Nazarov, S. Petermichl, S. Treil and A. Volberg [25] established the following quantitative estimate for W ∈ A 2 ,…”

Section: Introductionmentioning

confidence: 99%

Sharp $$A_{1}$$ Weighted Estimates for Vector-Valued Operators

Isralowitz

Pott²,

Rivera-Rı́os³

2020

J Geom Anal

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“…. This result was improved by Nazarov, et al [26] and Culiuc, di Plinio and Ou [11] and extended it to all Calderón-Zygmund operators T , [26] they prove a stronger result which we will discuss below.) Corollary 1.16 reduces to this estimate when p = 2.…”

Section: Introductionmentioning

confidence: 62%

“…16. In our proof we use the recent result of Nazarov, et al [26], who extended dyadic approximation theory for singular integrals to the matrix setting, and showed that to prove matrix weighted estimates for Calderón-Zygmund operators it is enough to prove them for sparse operators.…”

Section: Introductionmentioning

confidence: 99%

Two Weight Bump Conditions for Matrix Weights

Ofs

Isralowitz

Moen

2018

Integr. Equ. Oper. Theory

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In this paper we extend the theory of two weight, A p bump conditions to the setting of matrix weights. We prove two matrix weight inequalities for fractional maximal operators, fractional and singular integrals, sparse operators and averaging operators. As applications we prove quantitative, one weight estimates, in terms of the matrix A p constant, for singular integrals, and prove a Poincaré inequality related to those that appear in the study of degenerate elliptic PDEs.2010 Mathematics Subject Classification. Primary 42B20, 42B25, 42B35.

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“…The classical sharp scalar weighted result by Huković-Treil-Volberg [7] was proved via Bellman functions and only required the simple Carleson Lemma, not the bilinear version. The estimate for the Hilbert transform is still open, with best to date estimate by Nazarov-Petermichl-Treil-Volberg [10], missing the sharp conjecture by a half power of the A 2 characteristic. For most known operator norms, the question of sharpness is unsettled, but there usually is just a raised power on the dependence of the A 2 charateristic, not complete failure of the estimate, such as what we see in the case of BET.…”

Section: Introductionmentioning

confidence: 99%

Failure of the matrix weighted bilinear Carleson embedding theorem

Domelevo

Petermichl

Škreb

2019

Linear Algebra and its Applications

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We prove failure of the natural formulation of a matrix weighted bilinear Carleson embedding theorem, featuring a matrix valued Carleson sequence as well as products of norms for the embedding. We show that assuming an A 2 weight is also not sufficient. Indeed, a uniform bound on the conditioning number of the matrix weight is necessary and sufficient to get the bilinear embedding. We prove the optimal dependence of the embedding on this quantity. We show that any improvement of a recent matrix weighted bilinear embedding, featuring a scalar Carleson sequence and inner products instead of norms must fail. In particular, replacing the scalar sequence by a matrix sequence results in failure even when maintaining the formulation using inner products. Any formulation using norms, even in the presence of a scalar Carleson sequence must fail. As a positive result, we prove the so-called matrix weighted redundancy condition in full generality.

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Convex body domination and weighted estimates with matrix weights

Cited by 48 publications

References 18 publications

Sharp $$A_{1}$$ Weighted Estimates for Vector-Valued Operators

Sharp $$A_{1}$$ Weighted Estimates for Vector-Valued Operators

Two Weight Bump Conditions for Matrix Weights

Failure of the matrix weighted bilinear Carleson embedding theorem

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