Sparse Domination Theorem for Multilinear Singular Integral Operators with Lr-Hörmander Condition

Li, Kangwei

doi:10.1307/mmj/1516330973

Cited by 34 publications

(38 citation statements)

References 31 publications

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“…Therefore, assuming additionally that T is of weak type (q, q), by Corollary 5.1 we obtain that (5.1) holds with s = max(q, r ′ ). This result in a slightly different form can be found in [21]. Example 5.3.…”

Section: Examplesmentioning

confidence: 66%

“…Indeed, it is easy to see that f s,Q appears in the sparse domination estimate in Theorem 3.1 just because, by Hölder's inequality, We also note that Theorem 3.1 can be easily extended to a multilinear case. In [21], a multilinear extension of Theorem A was obtained. Our multilinear variant of Theorem 3.1 improves this result exactly in the same way as Theorem 3.1 improves Theorem A.…”

Section: Some Variations Of Theorem 11mentioning

confidence: 99%

“…Note that in Theorem 3.4, similarly to the corresponding result in [21], we do not assume that T is multilinear (or multi(sub)linear), which is in contrast to the statement of its linear analogue, Theorem 3.1. The explanation is in the way we defined M # T,α in the linear and multilinear cases.…”

Section: Some Variations Of Theorem 11mentioning

confidence: 99%

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Some Remarks on the Pointwise Sparse Domination

Lerner

Ombrosi

2019

J Geom Anal

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We obtain an improved version of the pointwise sparse domination principle established by the first author in [19]. This allows us to determine nearly minimal assumptions on a singular integral operator T for which it admits a sparse domination.where f p p,Q = 1 |Q| Q |f | p , and S is a sparse family of cubes of R n . Recall that the family of cubes S is called η-sparse, 0 < η ≤ 1, if for every cube Q ∈ S, there exists a measurable set E Q ⊂ Q such that |E Q | ≥ η|Q|, and the sets {E Q } Q∈S are pairwise disjoint.Localization and sparseness are two main ingredients which make sparse bounds especially effective in quantitative weighted norm inequalities.The literature about sparse bounds is too extensive to be given here in more or less adequate form. We mention only that sparse bounds for Calderón-Zygmund operators can be found in [7,15,16,18,19,20]. Also, there are several general sparse domination principles [5,6,8,19].In [19], a sparse domination principle was obtained in terms of the grand maximal truncated operatordefined for a given operator T .2010 Mathematics Subject Classification. 42B20, 42B25.

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Section: Examplesmentioning

confidence: 66%

Section: Some Variations Of Theorem 11mentioning

confidence: 99%

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Some Remarks on the Pointwise Sparse Domination

Lerner

Ombrosi

2019

J Geom Anal

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“…, we get as in the scalar-valued case (and with the same classical Calderón-Zygmund decomposition proof) that T : L 1 (R n ; E 1 ) × L 1 (R n ; E 2 ) → L 1/2,∞ (R n ; E 3 ). Such a weak type estimate implies a pointwise sparse domination -see for instance [47] for a proof in the scalar case. However, essentially the same proof works in the Banach-valued case.…”

Section: 4mentioning

confidence: 99%

Bilinear Calderón–Zygmund theory on product spaces

Martikainen

Vuorinen

2020

Journal de Mathématiques Pures et Appliquées

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We develop a wide general theory of bilinear bi-parameter singular integrals T . First, we prove a dyadic representation theorem starting from T 1 assumptions and apply it to show many estimates, including L p × L q → L r estimates in the full natural range together with weighted estimates and mixed-norm estimates. Second, we develop commutator decompositions and show estimates in the full range for commutators and iterated commutators, like [b1, T ]1 and [b2, [b1, T ]1]2, where b1 and b2 are little BMO functions. Our proof method can be used to simplify and improve linear commutator proofs, even in the two-weight Bloom setting. We also prove commutator lower bounds by using and developing the recent median method.

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“…with Q κ = 3 κ Q. This operator was introduced by Lerner [21] and plays an important role in the proof of weighted estimates for singular integral operators, see [24,4,25].…”

Section: Proof Of Theorem 14mentioning

confidence: 99%