Weak and Strong Type A 1 - A ∞ Estimates for Sparsely Dominated Operators.

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2019-01-01 Epub Date: 2018-02-06 DOI:10.1007/s12220-018-9989-2

Dorothee Frey, Zoe Nieraeth

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引用次数: 18

Abstract

We consider operators T satisfying a sparse domination property $\begin{matrix} | ⟨ T f, g ⟩ | \leq c \sum_{Q \in S} {⟨ f ⟩}_{p_{0}, Q} {⟨ g ⟩}_{q_{0}^{'}, Q} | Q | \end{matrix}$ with averaging exponents $1 \leq p_{0} < q_{0} \leq \infty$ . We prove weighted strong type boundedness for $p_{0} < p < q_{0}$ and use new techniques to prove weighted weak type $(p_{0}, p_{0})$ boundedness with quantitative mixed $A_{1}$ - $A_{\infty}$ estimates, generalizing results of Lerner, Ombrosi, and Pérez and Hytönen and Pérez. Even in the case $p_{0} = 1$ we improve upon their results as we do not make use of a Hörmander condition of the operator T. Moreover, we also establish a dual weak type $(q_{0}^{'}, q_{0}^{'})$ estimate. In a last part, we give a result on the optimality of the weighted strong type bounds including those previously obtained by Bernicot, Frey, and Petermichl.

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稀疏支配算子的弱和强A 1 - A∞估计。

我们考虑算子T满足一个稀疏支配性质|⟨T f, g⟩|≤c∑Q∈S⟨f⟩p 0, Q⟨g⟩Q 0 '， Q | Q |平均指数1≤p 0 Q 0≤∞。我们证明了p 0 p q 0的加权强型有界性，并利用新技术用定量混合a1 - A∞估计证明了加权弱型(p 0, p 0)有界性，推广了Lerner, Ombrosi, and p兼并和Hytönen and p兼并的结果。即使在p 0 = 1的情况下，我们也改进了他们的结果，因为我们没有利用算子t的Hörmander条件。此外，我们还建立了一个对偶弱类型(q0 '， q0 ')估计。在最后一部分中，我们给出了加权强类型界的最优性的结果，包括以前由Bernicot, Frey和Petermichl得到的结果。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.