A characterization of BMO in terms of endpoint bounds for commutators of singular integrals

Accomazzo, Natalia

doi:10.1007/s11856-018-1778-x

Cited by 10 publications

(13 citation statements)

References 15 publications

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“…It is well-known that the commutator of the Hilbert transform has the following properties: a) [ For more details, we refer to the references listed above. We also point out that there are quite a number of recent results on the characterisations of commutators in the above forms for singular integrals in different settings, see for example [10,22,9,21,25,24,15,19,13,8,1]. Inspired by these classical results above, it is natural to ask whether these results hold on the Heisenberg group H n .…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 85%

“…We also point out that this endpoint characterisation in Theorem 1.3 above is sharp since following the method in [24] in the Euclidean setting and using the lower bound in Theorem 1.1, it is easy to construct a function b ∈ BMO(G) such that [b, R j ] fails to be weak type (1,1).…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 89%

“…For the sufficiency, we point out that it follows directly from [24] with only minor changes since for this part we only need to use the upper bound of the Riesz transform, which satisfies the standard size and smoothness condition of Calderón-Zygmund type. For the necessary part, we use the idea of Uchiyama [27] and the technique that has been further explored and studied in [13] and [1]. To be more specific, we write |[b, R j ](f )(g)| ≥ |R j (bf )(g)| − |b(g)| |R j (f )(g)|, and then by choosing a suitable function f that is closely related to b and with cancellation condition and by making good use of the lower bound in Theorem 1.1, we show that |R j (bf )(g)| is the main term and |b(g)| |R j (f )(g)| acts as the "error" term due to the cancellation of f and the smoothness condition of the kernel of R j .…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

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Lower bound of Riesz transform kernels and commutator theorems on stratified nilpotent Lie groups

Duong

et al. 2019

Journal de Mathématiques Pures et Appliquées

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We provide a study of the Riesz transforms on stratified nilpotent Lie groups, and obtain a certain version of the pointwise lower bound of the Riesz transform kernel. Then we establish the characterisation of the BMO space on stratified nilpotent Lie groups via the boundedness of the commutator of the Riesz transforms and the BMO function. This extends the well-known Coifman, Rochberg, Weiss theorem from Euclidean space to the setting of stratified nilpotent Lie groups. In particular, these results apply to the wellknown example of the Heisenberg group. As an application, we also study the curl operator on the Heisenberg group and stratified nilpotent Lie groups, and establish the div-curl lemma with respect to the Hardy space on stratified nilpotent Lie groups.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 85%

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 89%

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Lower bound of Riesz transform kernels and commutator theorems on stratified nilpotent Lie groups

Duong

et al. 2019

Journal de Mathématiques Pures et Appliquées

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“…Observe that (1.4) contains as a particular case the endpoint estimate obtained in [14] and provides precise quantitative bound for the case in which the symbol has better local decay properties than BM O functions. We recall that in [1], it was shown that if a commutator of a certain singular integral satisfies a weak-type (1, 1) estimate then b ∈ L ∞ and that the L log L estimate, first settled in [31], implies that b ∈ BM O. Bearing those results in mind we wonder whether b ∈ Osc exp L r should be a neccesary condition for (1.4), at least in the case u = 1, to hold.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A sparse approach to mixed weak type inequalities

Caldarelli

Rivera-Rı́os

2019

Math. Z.

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In this paper we provide some quantitative mixed-type estimates assuming conditions that imply that uv ∈ A∞ for Calderón-Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced in [8] and extended in [23] and [25]Sawyer also conjectured that (1.1) should hold as well for the Hilbert transform. Cruz-Uribe, Martell and Pérez [7] generalized (1.1) to higher dimensions and actually proved that Sawyer's conjecture holds for Calderón-Zygmund operators via the following extrapolation argument.Theorem C. Assume that for every w ∈ A ∞ and some 0 < p < ∞,Then for every u ∈ A 1 and everyThe conditions on the weights in that extrapolation result lead them to conjecture that (1.1), and consequently the corresponding estimate for Calderón-Zygmund operators should hold as well with u ∈ A 1 and v ∈ A ∞ . That conjecture was positively answered recently in [24] where several quantitative estimates were provided as well. At this point we would like to mention, as well, a recent generalization provided for Orlicz maximal operators in [2].In [7], besides the aforementioned results, it was shown that (1.1) holds if u ∈ A 1 and v ∈ A ∞ (u) (see Section 2.2 for the precise definition of A p (u)). The advantage of that condition is that the product uv is an A ∞ weight. Over the past few years, there have been new contributions under those assumptions such as [3] for the case of fractional integrals and related operators, [28,29] for related quantitative estimates and [26] for multilinear extensions.

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“…Very recently we have learned that after finishing this paper some results concerning the necessity of BM O for the endpoint estimate of commutators have been obtained. We remit the interested reader to [1,9] for those results. In an even more recent work [14], a more general version of the second part of Theorem 1.1 has been obtained, answering positively a question posed in Remark 4.1.…”

Section: Remarks and Complementsmentioning

confidence: 96%