Sparse bilinear forms for Bochner Riesz multipliers and applications

Benea, Cristina; Bernicot, Frédéric; Luque, Teresa

doi:10.1112/tlm3.12005

Cited by 29 publications

(52 citation statements)

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“…Indeed when a " 1, b " pn`1q{2`δ, the kernels K a,b are the convolution kernels associated to the Bochner-Riesz multipliers m δ . Sparse bounds for Bochner-Riesz multipliers have been recently obtained by Lacey, Mena and Reguera in [33]; see also the endpoint result of Kesler and Lacey [27] or the previous work of Benea, Bernicot and Luque [4]. Besides the weighted A p estimates obtained as a consequence of the sparse domination, one should note that there are weighted Fefferman-Stein inequalities for Bochner-Riesz multipliers involving Kakeya-type maximal functions; see for instance the earlier work of Carbery [8] or Carbery and Seeger [10].…”

Section: 1mentioning

confidence: 76%

Sparse bounds for pseudodifferential operators

Beltran

Cladek

2020

JAMA

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We prove sparse bounds for pseudodifferential operators associated to Hörmander symbol classes. Our sparse bounds are sharp up to the endpoint and rely on a single scale analysis. As a consequence, we deduce a range of weighted estimates for pseudodifferential operators. The results naturally apply to the context of oscillatory Fourier multipliers, with applications to dispersive equations and oscillatory convolution kernels.2010 Mathematics Subject Classification. Primary: 35S05, Secondary: 42B25. Key words and phrases. Pseudodifferential operators, sparse domination, weighted theory. 1 We use the notation A À B to denote that there is a constant C such that A ď CB and the notation A Àǫ B to specify the dependence of the implicit constant in a certain parameter ǫ. We omit the constant factors of π coming from our normalisation of the Fourier transform. p p,Q :" |Q|´1 ş Q |f | p dx. Given 1 ď r, s ă 8, the pr, sq sparse form Λ S,r,s is defined as Λ S,r,s pf, gq :" ÿ QPS |Q| f r,Q g s,Q , and the r-sparse operator A r,S is defined as A r,S f pxq :" ÿ QPS f r,Q χ Q pxq. Theorem 1.2. Let a P S m ρ,δ for some m ă 0 and 0 ă δ ď ρ ă 1. Then for any compactly supported bounded functions f, g on R n , there exist sparse collections S and r S of dyadic cubes such that | T a f, g | ď Cpm, ρ, r, sqΛ S,r,s 1 pf, gqand | T a f, g | ď Cpm, ρ, r, sqΛ r S,s 1 ,r pf, gq for all pairs pr, s 1 q and ps 1 , rq such that m ă´np1´ρqp1{r´1{2q, 1 ď r ď s ď 2 2 Given 1 ď r ď 8, r 1 denotes its conjugate Hölder exponent, that is 1{r`1`r 1 " 1.

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Section: 1mentioning

confidence: 76%

Sparse bounds for pseudodifferential operators

Beltran

Cladek

2020

JAMA

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“…Our formulation in terms of positive sparse forms overcomes this obstacle: a similar idea, albeit not explicit, appears in the linear setting in . After the first version of this article was made public, several works based on sparse form domination have appeared within and beyond Calderón‐‐Zygmund theory, see for example and references therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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 far as we know, the only vector-valued estimates that have been shown for B δ have been for X = ℓ s , see [5]. For any p − , p + and δ for which B δ S(p − ,p + ) < ∞, we obtain by Theorem 5.5 that inequality (5.3) with T = B δ holds for any Banach function space X satisfying X ∈ UMD p − ,p + , yielding new vector-valued estimates.…”

Section: Multilinear Calderón-zygmund Operatorsmentioning

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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Sparse bilinear forms for Bochner Riesz multipliers and applications

Abstract: We use the very recent approach developed by Lacey in [23] and extended by Bernicot-Frey-Petermichl in [3], in order to control Bochner-Riesz operators by a sparse bilinear form. In this way, new quantitative weighted estimates, as well as vector-valued inequalities are deduced

Cited by 29 publications

References 40 publications

Sparse bounds for pseudodifferential operators

Sparse bounds for pseudodifferential operators

Domination of multilinear singular integrals by positive sparse forms

Vector-Valued Extensions of Operators Through Multilinear Limited Range Extrapolation

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