Multiple vector-valued inequalities via the helicoidal method

Benea, Cristina; Muscalu, Camil

doi:10.2140/apde.2016.9.1931

Cited by 49 publications

(182 citation statements)

References 35 publications

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“…As a further application of Corollary , weighted, vector‐valued estimates for multipliers

T_{m}

satisfying condition , extending the results of can be obtained by a multilinear version of the extrapolation theory of . These extensions are the object of an upcoming companion article by the same authors.…”

Section: Introduction and Main Resultsmentioning

confidence: 88%

“…For each pair (q 1 , q 2 ), the range of tuples r for which T m admits L q1 × L q2 a bounded vectorvalued extension is the same as the one recently obtained in [3,Theorem 7] for the vector-valued bilinear Hilbert transforms. Condition (A.5) needs to be imposed in order to ensure that the set { p = (p 1 , p 2 , p 3 ) open admissible : p j < min{r j , q j }, j = 1, 2, 3.}…”

Section: Handling the Tail Termsmentioning

confidence: 96%

“…Throughout this proof only, we make an exception concerning our use of the almost inequality sign: the implied constant is allowed to be of the form K(A N ) 3 where K and N depend only on the tuple p. For the sake of brevity, we assume that all functions f j are supported on 3Q. With reference to (5.1) for I f, p,Q , let…”

Section: Proof Of Lemma 51mentioning

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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“…As a further application of Corollary , weighted, vector‐valued estimates for multipliers

T_{m}

Section: Introduction and Main Resultsmentioning

confidence: 88%

Section: Handling the Tail Termsmentioning

confidence: 96%

See 1 more Smart Citation

Domination of multilinear singular integrals by positive sparse forms

Culiuc

Plinio

2018

J. London Math. Soc.

107

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“…Several extensions and refinements of [30] have since appeared, see e.g. [1,4,8,9]. In general, as Corollary 1.2 demonstrates, Theorem 1.1 is outside the scope of the above references, although it does imply a strict subset of the ℓ p estimates of [30].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Banach-valued multilinear singular integrals

Plinio¹,

Ou²

2018

Indiana Univ. Math. J.

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We prove that the class of trilinear multiplier forms with singularity over a one dimensional subspace, including the bilinear Hilbert transform, admit bounded L pextension to triples of intermediate UMD spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of UMD spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the UMD-valued setting. This is then employed to obtain appropriate single tree estimates by appealing to the UMD-valued bound for bilinear Calderón-Zygmund operators recently obtained by the same authors.

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“…In Section 4, we present a new application which concerns mixed-norm estimates (a topical subject, see e.g. [2,3] and [14]). Here we work with tensor products T n ⊗T m , where T n and T m are bilinear Calderón-Zygmund operators in R n and R m .…”

Section: Introductionmentioning

confidence: 99%

End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications

Martell

Martikainen

et al. 2020

Trans. Amer. Math. Soc.

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In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here we consider the situations where some of the exponents of the Lebesgue spaces appearing in the hypotheses and/or in the conclusion can be possibly infinity. The scheme we follow is similar, but, in doing that, we need to develop a one-variable end-point off-diagonal extrapolation result. This complements the corresponding "finite" case obtained by Duoandikoetxea, which was one of the main tools in the aforementioned paper. The second goal of this paper is to present some applications. For example, we obtain the full range of mixed-norm estimates for tensor products of bilinear Calderón-Zygmund operators with a proof based on extrapolation and on some estimates with weights in some mixed-norm class. The same occurs with the multilinear Calderón-Zygmund operators, the bilinear Hilbert transform, and the corresponding commutators with BMO functions. Extrapolation along with the already established weighted norm inequalities easily give scalar and vector-valued inequalities with multilinear weights and these include the end-point cases.

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Multiple vector-valued inequalities via the helicoidal method

Cited by 49 publications

References 35 publications

Domination of multilinear singular integrals by positive sparse forms

Domination of multilinear singular integrals by positive sparse forms

Banach-valued multilinear singular integrals

End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications

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