Quantitative estimates and extrapolation for multilinear weight classes

Nieraeth, Zoe

doi:10.1007/s00208-019-01816-5

Cited by 39 publications

(69 citation statements)

References 34 publications

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“…To conclude with this introduction we would like to mention that some interesting related work has recently appeared while this manuscript was in preparation. Nieraeth in [31] has obtained a result similar to ours with a proof which uses an independent different method. In a nutshell, Theorem 2.1 -which was announced in [27] before [31] was posted-is proved by following [27, Proof of Theorem 1.1] with some appropriate changes both in the argument and in the notation.…”

Section: Introductionsupporting

confidence: 76%

“…Nieraeth in [31] has obtained a result similar to ours with a proof which uses an independent different method. In a nutshell, Theorem 2.1 -which was announced in [27] before [31] was posted-is proved by following [27, Proof of Theorem 1.1] with some appropriate changes both in the argument and in the notation. Having said that, the main tool that we need here is Theorem 2.3, which extends [16,Theorem 5.1] (and also [23]) to allow for end-point estimates.…”

Section: Introductionsupporting

confidence: 76%

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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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Section: Introductionsupporting

confidence: 76%

Section: Introductionsupporting

confidence: 76%

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

Martell

Martikainen

et al. 2020

Trans. Amer. Math. Soc.

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“…Remark 1.3. In the recent work [53] of Li, Martell and Ombrosi, and the recent work [58] of the second author, scalar-valued extrapolation results were obtained using the multilinear weight classes from [52], which were made public after this paper first appeared. Rather than considering a condition for each weight individually, these weight classes allow for an interaction between the various weights, making them more appropriately adapted to the multilinear setting.…”

Section: Introductionmentioning

confidence: 99%

Vector-Valued Extensions of Operators Through Multilinear Limited Range Extrapolation

Lorist

Nieraeth

2019

J Fourier Anal Appl

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We give an extension of Rubio de Francia's extrapolation theorem for functions taking values in UMD Banach function spaces to the multilinear limited range setting. In particular we show how boundedness of an m-(sub)linear operatorfor a certain class of Muckenhoupt weights yields an extension of the operator to Bochner spaces L p (w p ; X) for a wide class of Banach function spaces X, which includes certain Lebesgue, Lorentz and Orlicz spaces.We apply the extrapolation result to various operators, which yields new vector-valued bounds. Our examples include the bilinear Hilbert transform, certain Fourier multipliers and various operators satisfying sparse domination results.2010 Mathematics Subject Classification. Primary: 42B25; Secondary: 42B15, 46E30.Vector-valued extensions of the extrapolation theory have also been considered. Through an argument using Fubini's Theorem, the initial estimate (1.1) immediately implies not only the estimate (1.2) for all p ∈ (1, ∞), but also for extensions of the operator T to functions taking values in the sequence spaces ℓ s or more generally Lebesgue spaces L s for s ∈ (1, ∞). Moreover, Rubio de Francia showed in [60, Theorem 5] that one can take this even further. Indeed, this result states that assuming (1.1) holds for some p 0 ∈ (1, ∞) and for all weights w ∈ A p 0 , then for each Banach function space X with the UMD property, T extends to an operator T on the Bochner space L p (X) which satisfiesfor all p ∈ (1, ∞) and all f ∈ L p (X) . Recently, it was shown by Amenta, Veraar, and the first author in [2] that given p − ∈ (0, ∞), if (1.1) holds for p 0 ∈ (p − , ∞) and all weights w ∈ A p 0 /p − , then for each Banach function space X such that X p − has the UMD property, T extends to an operator T on the Bochner space L p (w; X) and satisfiesfor all p ∈ (p − , ∞), all weights w ∈ A p/p − and all f ∈ L p (w; X). Here X p − is the p − -concavification of X, see Section 2 for the definition.Vector-valued estimates in harmonic analysis have been actively developed in the past decades. Important for the mentioned vector-valued extrapolation are the equivalence of the boundedness of the vector-valued Hilbert transform on L p (X) and the UMD property of X for a Banach space X (see [9,13]) and the fact that for a Banach function space X the UMD property implies the boundedness of the lattice Hardy-Littlewood maximal operator on L p (X) (see [10,61]). For recent results in vector-valued harmonic analysis in UMD Banach function spaces, see for example [8,25,36,41,66].

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“…For p 3 ≤ 1 it follows from the sparse domination for BHF Π established in Theorem 8.4: this implies weighted estimates which, by extrapolation, yield the claimed bounds (97) (plus further weighted estimates that we do not discuss here). The details of this weighted multilinear extrapolation are worked out in [38,Section 4.1] (see also [29,30]).…”

Section: Bounds For Bht Including Quasi-banach Exponentsmentioning

confidence: 99%

“…Benea and Muscalu [4,5] extended this result to mixednorm spaces, including L ∞ and some quasi-Banach spaces, by a new 'helicoidal' method. 5 The multilinear extrapolation theorems of Lorist and Nieraeth [32,33,38] allow for even more general function spaces, using the weighted scalar-valued bounds of [12,13] or the sparse domination proved in [6,13].…”

Section: Introductionmentioning

confidence: 99%

The bilinear Hilbert transform in UMD spaces

Amenta

Uraltsev

2020

Math. Ann.

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We prove $$L^p$$ L p -bounds for the bilinear Hilbert transform acting on functions valued in intermediate UMD spaces. Such bounds were previously unknown for UMD spaces that are not Banach lattices. Our proof relies on bounds on embeddings from Bochner spaces $$L^p(\mathbb {R};X)$$ L p ( R ; X ) into outer Lebesgue spaces on the time-frequency-scale space $$\mathbb {R}^3_+$$ R + 3 .

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Quantitative estimates and extrapolation for multilinear weight classes

Cited by 39 publications

References 34 publications

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

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

Vector-Valued Extensions of Operators Through Multilinear Limited Range Extrapolation

The bilinear Hilbert transform in UMD spaces

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