Intuitive dyadic calculus: The basics

Lerner, Andrei K.; Nazarov, Fëdor

doi:10.1016/j.exmath.2018.01.001

Cited by 168 publications

(190 citation statements)

References 12 publications

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“…Such a pointwise domination principle, albeit in a slightly weaker sense, appears explicitly for the first time in the proof of the

A_{2}

theorem by Lerner . We also point out the recent improvements by Lacey and Lerner , and the analogue for multilinear Calderón‐‐Zygmund operators obtained independently by Conde‐Alonso and Rey and by Lerner and Nazarov . Most recently, Bernicot, Frey and Petermichl extend this approach to nonintegral singular operators associated with a second‐order elliptic operator, lying outside the scope of classical Calderón–Zygmund theory.…”

Section: Introduction and Main Resultsmentioning

confidence: 71%

“…Remark For bilinear Calderón‐‐Zygmund operators

T

, there holds a pointwise domination by sparse operators of the type

0true | T (f_{1}, f_{2}) (x) | ⩽ C \sum_{I \in scriptS false(f_{1}, f_{2} false)} {false⟨ f_{1} false⟩}_{I, p_{1}} {false⟨ f_{2} false⟩}_{I, p_{2}} 1_{I} (x) .

One can take

p_{1} = p_{2} = 1

: see . Essentially self‐adjoint operators

T

enjoying such pointwise domination inherit the boundedness property

T : L^{1} \times L^{p_{j}} \to L^{\frac{p_{j}}{1 + p_{j}}, \infty}

which, as described in the previous Remark , fails for the generic

T_{m}

of the class when

inf {p_{1}, p_{2}} < 2

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…One can take p 1 = p 2 = 1: see [23]. Essentially self-adjoint operators T enjoying such pointwise domination inherit the boundedness property…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Then

\begin{matrix} true \\ left {sans-serifPSF}_{S}^{\overset{p}{true}} (g_{1} w_{1}^{\frac{1}{p_{1}}}, g_{2} w_{2}^{\frac{1}{p_{2}}}, g_{3} w_{3}^{\frac{1}{p_{3}}}) = \sum_{Q \in S} false| Q false| \prod_{j = 1}^{3} {((), {⟨ g_{j}^{p_{j}} w_{j} ⟩}_{Q})}^{\frac{1}{p_{j}}} \\ left = \sum_{Q \in S} (0true \prod_{j = 1}^{3} w_{j} {(E_{Q})}^{\frac{1}{q_{j}}} {((), \frac{{false⟨ g_{j}^{p_{j}} w_{j} false⟩}_{Q}}{{false⟨ w_{j} false⟩}_{Q}})}^{\frac{1}{p_{j}}}) \times (0true \prod_{j = 1}^{3} {⟨ w_{j} ⟩}_{Q}^{\frac{1}{p_{j}} - \frac{1}{q_{j}}}) \times (0true false| Q false| \prod_{j = 1}^{3} {((), \frac{{false⟨ w_{j} false⟩}_{Q}}{w_{j} false(E_{Q} false)})}^{\frac{1}{q_{j}}}) . \end{matrix}

The second product inside the sum of is the precursor to

{false[\overset{v}{true} false]}_{A_{\vec{q}}^{\vec{p}}}

. Arguing as in , the rightmost factor in is bounded above uniformly in

Q<...>

…”

Section: Proof Of Theorem and Corollarymentioning

confidence: 99%

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Domination of multilinear singular integrals by positive sparse forms

Culiuc

Plinio

2018

J. London Math. Soc.

107

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We establish a uniform domination of the family of trilinear multiplier forms with singularity over a one‐dimensional subspace by positive sparse forms involving Lp‐averages. This class includes the adjoint forms to the bilinear Hilbert transforms. Our result strengthens the Lp‐boundedness proved by Muscalu, Tao and Thiele, and entails as a corollary a novel rich multilinear weighted theory. A particular case of this theory is the Lqfalse(v1false)×Lqfalse(v2false)‐boundedness of the bilinear Hilbert transform when the weight vj belong to the class Aq+12∩RH2. Our proof relies on a stopping time construction based on newly developed localized outer‐Lp embedding theorems for the wave packet transform. In the Appendix, we show how our domination principle can be applied to recover the vector‐valued bounds for the bilinear Hilbert transforms recently proved by Benea and Muscalu.

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“…Such a pointwise domination principle, albeit in a slightly weaker sense, appears explicitly for the first time in the proof of the

A_{2}

Section: Introduction and Main Resultsmentioning

confidence: 71%

“…Remark For bilinear Calderón‐‐Zygmund operators

T

, there holds a pointwise domination by sparse operators of the type

0true | T (f_{1}, f_{2}) (x) | ⩽ C \sum_{I \in scriptS false(f_{1}, f_{2} false)} {false⟨ f_{1} false⟩}_{I, p_{1}} {false⟨ f_{2} false⟩}_{I, p_{2}} 1_{I} (x) .

One can take

p_{1} = p_{2} = 1

: see . Essentially self‐adjoint operators

T

enjoying such pointwise domination inherit the boundedness property

T : L^{1} \times L^{p_{j}} \to L^{\frac{p_{j}}{1 + p_{j}}, \infty}

which, as described in the previous Remark , fails for the generic

T_{m}

of the class when

inf {p_{1}, p_{2}} < 2

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…One can take p 1 = p 2 = 1: see [23]. Essentially self-adjoint operators T enjoying such pointwise domination inherit the boundedness property…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Then

\begin{matrix} true \\ left {sans-serifPSF}_{S}^{\overset{p}{true}} (g_{1} w_{1}^{\frac{1}{p_{1}}}, g_{2} w_{2}^{\frac{1}{p_{2}}}, g_{3} w_{3}^{\frac{1}{p_{3}}}) = \sum_{Q \in S} false| Q false| \prod_{j = 1}^{3} {((), {⟨ g_{j}^{p_{j}} w_{j} ⟩}_{Q})}^{\frac{1}{p_{j}}} \\ left = \sum_{Q \in S} (0true \prod_{j = 1}^{3} w_{j} {(E_{Q})}^{\frac{1}{q_{j}}} {((), \frac{{false⟨ g_{j}^{p_{j}} w_{j} false⟩}_{Q}}{{false⟨ w_{j} false⟩}_{Q}})}^{\frac{1}{p_{j}}}) \times (0true \prod_{j = 1}^{3} {⟨ w_{j} ⟩}_{Q}^{\frac{1}{p_{j}} - \frac{1}{q_{j}}}) \times (0true false| Q false| \prod_{j = 1}^{3} {((), \frac{{false⟨ w_{j} false⟩}_{Q}}{w_{j} false(E_{Q} false)})}^{\frac{1}{q_{j}}}) . \end{matrix}

The second product inside the sum of is the precursor to

{false[\overset{v}{true} false]}_{A_{\vec{q}}^{\vec{p}}}

. Arguing as in , the rightmost factor in is bounded above uniformly in

Q<...>

…”

Section: Proof Of Theorem and Corollarymentioning

confidence: 99%

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Domination of multilinear singular integrals by positive sparse forms

Culiuc

Plinio

2018

J. London Math. Soc.

107

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“…The generation of the dyadic basis, from [14] The most important features of our basis of dyadic cubes are as follows:…”

Section: The Geometry Of the Dyadic Cubesmentioning

confidence: 99%

On the harmonic and geometric maximal operators

Duffee¹

2018

Mathematical Inequalities &Amp; Applications

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In this dissertation, we attempt to characterize the boundedness of two operators over certain spaces of functions. The operators in question, the harmonic maximal operator and the geometric maximal operator, arise naturally from consideration of the paradigmatic maximal operator, the Hardy-Littlewood maximal function, and from consideration of well-known analogues of the simple arithmetic mean. In particular, we seek to improve upon earlier work by removing certain unwieldy assumptions, thus moving closer to a complete characterization of the L p -boundedness of both our operators for a pair of weights.This work primarily concerns weighted norm inequalities for two operators, the harmonic maximal operator and geometric maximal operator, given either a dyadic basis or a general basis. For any basis of cubes Q, we define those operators as follows. The harmonic maximal operator,and the geometric maximal operatorWe begin by considering our two operators restricted to the dyadic basis. The geometry of this basis allows us to decompose and recompose integrals of our operators taken over R n , a kind of discretization that simplifies the methods necessary for characterization. Within this framework we are able to remove the technical obstacles that ii lead to the unwieldy doubling conditions found in previous work.Having achieved a complete characterization in our well-behaved dyadic basis, we turn to a more difficult situation: that of a general basis. The principles necessary to treat our maximal operators in the standard basis of cubes or the dyadic basis can be abstracted further, leading to more general results that apply to a large extent to any measure regardless of its geometry.We conclude by investigating the necessary geometric theory to recover the standard basis results from our dyadic results.iii

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Mixed weak‐type inequalities in Euclidean spaces and in spaces of the homogeneous type

Ibañez‐Firnkorn,

Rivera‐Ríos

2023

Mathematische Nachrichten

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In this paper, we provide mixed weak‐type inequalities generalizing previous results in an earlier work by Caldarelli and the second author and also in the spirit of earlier results by Lorente et al. One of the main novelties is that, besides obtaining estimates in the Euclidean setting, results are provided as well in spaces of the homogeneous type, being the first mixed weak‐type estimates that we are aware of in that setting.

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Intuitive dyadic calculus: The basics

Cited by 168 publications

References 12 publications

Domination of multilinear singular integrals by positive sparse forms

Domination of multilinear singular integrals by positive sparse forms

On the harmonic and geometric maximal operators

Mixed weak‐type inequalities in Euclidean spaces and in spaces of the homogeneous type

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