𝐿^{𝑝} theory for outer measures and two themes of Lennart Carleson united

Do, Yen; Thiele, Christoph

doi:10.1090/s0273-0979-2014-01474-0

Cited by 47 publications

(161 citation statements)

References 20 publications

(24 reference statements)

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“…In fact, while Proposition is proved here via a transference argument based upon [, Theorem 1], a direct proof can be given by repeating the arguments of in the discrete setting. The construction of the outer

L^{p}

spaces on rank 1 collections, which parallels the outer

L^{p}

theory introduced by Do and Thiele in , is performed in Section 3.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Proof Define another size

0true s^{1} (F) (T) : = \frac{1}{| I_{T} |} \sum_{P \in sans-serifT} | I_{P} | | F false(P false) | .

Then it is obvious that for any

sans-serifT

there holds

0true \sum_{P \in sans-serifT} | I_{P} | \prod_{j = 1}^{3} | G_{j} false(P false) | ⩽ | I_{T} | s^{1} ((), \prod_{j = 1}^{3} G_{j}) (T),

which by the Radon–Nikodym proposition [, Proposition 3.6] implies

0true \sum_{P \in double-struckP} | I_{P} | \prod_{j = 1}^{3} | G_{j} false(P false) | ≲ {‖0true \prod_{j = 1}^{3} G_{j}‖}_{L^{1} (P, {sans-serifs}^{1})} .

Furthermore, according to and the classical Hölder's inequality, one can easily check that for any fixed

sans-serifT

\begin{matrix} true \\ left s^{1} ((), \prod) \end{matrix}

…”

Section: Outer Lp Spaces Of Tritilesmentioning

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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L^{p}

spaces on rank 1 collections, which parallels the outer

L^{p}

theory introduced by Do and Thiele in , is performed in Section 3.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Proof Define another size

0true s^{1} (F) (T) : = \frac{1}{| I_{T} |} \sum_{P \in sans-serifT} | I_{P} | | F false(P false) | .

Then it is obvious that for any

sans-serifT

there holds

0true \sum_{P \in sans-serifT} | I_{P} | \prod_{j = 1}^{3} | G_{j} false(P false) | ⩽ | I_{T} | s^{1} ((), \prod_{j = 1}^{3} G_{j}) (T),

which by the Radon–Nikodym proposition [, Proposition 3.6] implies

0true \sum_{P \in double-struckP} | I_{P} | \prod_{j = 1}^{3} | G_{j} false(P false) | ≲ {‖0true \prod_{j = 1}^{3} G_{j}‖}_{L^{1} (P, {sans-serifs}^{1})} .

Furthermore, according to and the classical Hölder's inequality, one can easily check that for any fixed

sans-serifT

\begin{matrix} true \\ left s^{1} ((), \prod) \end{matrix}

…”

Section: Outer Lp Spaces Of Tritilesmentioning

confidence: 99%

Domination of multilinear singular integrals by positive sparse forms

Culiuc

Plinio

2018

J. London Math. Soc.

107

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“…Then The following Lemma generalizes [6,Proposition 3.5]. Below pX, S, µq and pY, S 1 , νq are given outer measure spaces.…”

Section: Some Background On Outer Measure Spacesmentioning

confidence: 87%

“…Many standard properties of classical L p spaces can be proved for outer L p spaces, see [6] for details. We summarize several estimates from [6].…”

Section: Some Background On Outer Measure Spacesmentioning

confidence: 99%

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Variational estimates for the bilinear iterated Fourier integral

Muscalu

Thiele³

2017

Journal of Functional Analysis

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We prove pointwise variational L p bounds for a bilinear Fourier integral operator in a large but not necessarily sharp range of exponents. This result is a joint strengthening of the corresponding bounds for the classical Carleson operator, the bilinear Hilbert transform, the variation norm Carleson operator, and the bi-Carleson operator. Terry Lyon's rough path theory allows for extension of our result to multilinear estimates. We consider our result a proof of concept for a wider array of similar estimates with possible applications to ordinary differential equations.

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The ∞‐Besov capacity problem

Milman

Xiao

2017

Mathematische Nachrichten

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A theory of ∞-Besov capacities is developed and several applications are provided. In particular, we solve an open problem in the theory of limits of the ∞-Besov semi-norms, we obtain new restriction-extension inequalities, and we characterize the pointwise multipliers acting on the ∞-Besov spaces.where ∈ [1, ∞), −1 is the unit sphere of ℝ ≥1 , is the angle derivation from the vertical, and is the standard surface area measure. In [22]where ∈ [1, ∞), and −1 denotes the surface measure of −1 . A related inequality obtained in [5], and sharpened in [22], can be formulated asMathematische Nachrichten. 2017;290:2961-2976.

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𝐿^{𝑝} theory for outer measures and two themes of Lennart Carleson united

Cited by 47 publications

References 20 publications

Domination of multilinear singular integrals by positive sparse forms

Domination of multilinear singular integrals by positive sparse forms

Variational estimates for the bilinear iterated Fourier integral

The ∞‐Besov capacity problem

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