Exponential decay estimates for singular integral operators

Advances in Harmonic Analysis and Operator Theory

Rela

2013

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In this expository article we collect and discuss some recent results on different consequences of a Sharp Reverse Hölder Inequality for A∞ weights. For two given operators T and S, we study L p (w) bounds of Coifman-Fefferman type:, that can be understood as a way to control T by S.We will focus on a quantitative analysis of the constants involved and show that we can improve classical results regarding the dependence on the weight w in terms of Wilson's A∞ constantWe will also exhibit recent improvements on the problem of finding sharp constants for weighted norm inequalities involving several singular operators. In the same spirit as in [10], we obtain mixed A 1 -A∞ estimates for the commutator [b, T ] and for its higher order analogue T k b . A common ingredient in the proofs presented here is a recent improvement of the Reverse Hölder Inequality for A∞ weights involving Wilson's constant from [10].

Section: 3supporting

confidence: 67%

Section: 3mentioning

confidence: 99%

Improving Bounds for Singular Operators via Sharp Reverse Höolder Inequality for $$ A_\infty $$

Ortiz-Caraballo

Advances in Harmonic Analysis and Operator Theory

Rela

2013

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“…Denote b m = Q∈Sm χ Q , then A Sm f ≤ 2 m+1 b m . and therefore, if we denote S * m is the collection of maximal dyadic cubes in S m , taking into account the local exponential decay for sparse operators (see for instance [29]),…”

Section: Proof Of Theoremmentioning

confidence: 99%

Vector-valued operators, optimal weighted estimates and the Cp condition

Cejas

et al. 2020

Sci. China Math.

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In this paper some new results concerning the C p classes introduced by Muckenhoupt [28] and later extended by Sawyer [39], are provided. In particular we extend the result to the full range expected p > 0, to the weak norm, to other operators and to their vector-valued extensions. Some of those results rely upon sparse domination results that in some cases we provide as well. We will also provide sharp weighted estimates for vector valued extensions relying on those sparse domination results.

“…We use now one of the key estimates from [24]. Indeed, let {E(P )} P ∈S, P ⊆Q j be the family of sets from Definition 3.…”

Section: Definition 2 (Dyadic Lattice) a Dyadic Lattice D In R N Is mentioning

confidence: 99%

“…Motivated by this result of Karagulyan, Ortiz-Caraballo, Rela and the first author developed a new method for proving (6) in [24]. This method is flexible enough to deal with other operators including the commutators.…”

Section: Introductionmentioning

confidence: 99%

Three Observations on Commutators of Singular Integral Operators with BMO Functions

Association for Women in Mathematics Series

Rivera-Rı́os

2017

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Three observations on commutators of Singular Integral Operators with BMO functions are exposed, namely Section 2 The already known subgaussian local decay for the commutator, namelyis sharp, since it cannot be better than subgaussian. Section 3 It is not possible to obtain a pointwise control of the commutator by a finite sum of sparse operators defined by L log L averages. Section 4 Motivated by the conjugation method for commutators, it is shown the failure of the following endpoint estimate, if w ∈ A p \ A 1 then