Sparse Bounds and Improving Estimates, Continuous and Discrete

稀疏界限和改进估计，连续和离散

• 批准号: 1949206
• 负责人: Michael Lacey
• 金额: $ 29.77万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-06-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1949206&HistoricalAwards=false
• 关键词: Sparse; Bounds; Improving; Estimates; Continuous

Averaging always smooths out, or improves, functions or data. In the continuous case, this phenomena has been intensively studied for decades. If we average over objects in a discrete setting like the integers, or some other high dimensional lattice, the smoothing properties of the average have only recently started to be studied. A natural object to study is the average over a discrete sphere in a five dimensional lattice. The proof that the average improves functions engages a range of deep aspects within number theory and analysis. However, resolving the natural limits for this question still seem difficult. In this project, these questions will be explored in a setting that involves both continuous and discrete phenomena. These questions are also elementary to state, which makes them amenable for mentoring programs, from undergraduate through postdoctoral levels. An improving inequality in the continuous setting has been widely studied since the 1970's. However, the corresponding questions in the discrete setting have only just attracted attention. For instance, averages over the discrete sphere in the d-dimensional integers, with d at least 5, have a rich theory of improving inequalities, paralleling the much better known continuous case. The proofs however involve complications arising from multi-frequency analysis, as well as fine estimates on Kloosterman sums. There is a richer theory, the outlines of which are appearing, giving a sharp range of inequalities for some arithmetic varieties. This subject also allows for the proof of certain sparse bounds. The latter are scale-free versions of the improving inequalities. A sparse bound immediately implies other weighted and vector valued consequences. The latter are new in this subject. These questions reveal new aspects of these averaging operators, and require new modes of investigation, deepening the connection between harmonic analysis and number theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

求平均值总是平滑或改进函数或数据。 在连续的情况下，这种现象已经被深入研究了几十年。如果我们对离散设置中的对象（如整数或其他高维格）进行平均，则平均值的平滑特性直到最近才开始研究。 一个自然的研究对象是在一个五维格子离散球的平均。证明平均值可以改善函数涉及数论和分析中的一系列深层次问题。然而，解决这个问题的自然限制似乎仍然很困难。在这个项目中，这些问题将在涉及连续和离散现象的环境中进行探索。 这些问题也是基本的状态，这使得他们适合辅导计划，从本科到博士后水平。 自20世纪70年代以来，连续环境下的改进不等式得到了广泛的研究。然而，在离散设置中的相应问题才刚刚引起注意。例如，d维整数（d至少为5）的离散球面上的平均值有丰富的改进不等式的理论，类似于更好地了解连续的情况。 然而，证据涉及多频分析所产生的并发症，以及Kloosterman总和的精细估计。有一个更丰富的理论，其轮廓正在出现，为一些算术变种提供了一个尖锐的不等式范围。 这个主题也允许某些稀疏界限的证明。后者是改进的不平等的无标度版本。一个稀疏的界限立即意味着其他加权和向量值的后果。 后者在这个问题上是新的。这些问题揭示了这些平均算子的新方面，并需要新的调查模式，加深谐波分析和数论之间的联系。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

项目成果

Endpoint $\ell^r$ improving estimates for prime averages

端点 $ell^r$ 改进了素数平均值的估计

• DOI: 10.4310/mrl.2022.v29.n6.a6
• 发表时间: 2022
• 期刊: Mathematical Research Letters
• 影响因子: 1
• 作者: Lacey, Michael T.;Mousavi, Hamed;Rahimi, Yaghoub
• 通讯作者: Rahimi, Yaghoub

Compactness of commutator of Riesz transforms in the two weight setting

• DOI: 10.1016/j.jmaa.2021.125869
• 发表时间: 2020-10
• 期刊: Journal of Mathematical Analysis and Applications
• 影响因子: 1.3
• 作者: M. Lacey;Ji Li

Schatten Classes and Commutators of Riesz Transform on Heisenberg Group and Applications

• DOI: 10.1007/s00041-023-09995-1
• 发表时间: 2021-07
• 期刊: Journal of Fourier Analysis and Applications
• 影响因子: 1.2
• 作者: Zhijie Fan;Michael Lacey;Ji Li

On the convergence of multiple ergodic means

关于多重遍历均值的收敛性

• 发表时间: 2022
• 期刊: New York journal of mathematics
• 影响因子: 0.6
• 作者: Karagulyan, Grigori A.;Lacey, Michael T.;Martirosyan, Vahan A
• 通讯作者: Martirosyan, Vahan A

Commutators of Cauchy-Szego type integrals for domains in C^n with minimal smoothness

• DOI: 10.1512/iumj.2021.70.8573
• 发表时间: 2021
• 期刊: Indiana University Mathematics Journal
• 影响因子: 1.1
• 作者: X. Duong;M. Lacey;Ji Li;B. Wick;Qingyan Wu