Endpoint Behavior of Modulation Invariant Singular Integrals

调制不变奇异积分的端点行为

• 批准号: 1650810
• 负责人: Francesco DiPlinio
• 金额: $ 11.12万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-05-16 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1650810&HistoricalAwards=false
• 关键词: Endpoint; Behavior; Modulation; Invariant; Singular

Harmonic analysis studies how signals (functions) break up into a superposition of basic harmonics--signals with a well-specified duration, intensity and frequency--and how operations (filtering) applied to these components affect the reconstructed signal. Variants of this time-frequency decomposition process are performed in countless real-world applications, such as audio or image compression and filtering, image pattern recognition, data assimilation and denoising. One of the broad objectives of this project is the investigation of the theoretical feasibility threshold of the time-frequency techniques in terms of the relative size and smoothness of the input. An analogous procedure is adopted in tomographic imaging, where a solid body is reconstructed by means of sampling its density along penetrating waves, mathematically described as lines in three-dimensional space. This project will study mathematical toy models of sampling along lines or curves, whose theoretical understanding may play a significant role in the derivation of improved analytical image reconstruction methods. An integral component of the project is the training of graduate and undergraduate students within the active research group in harmonic analysis at Brown University, with the particular intent of attracting young and promising researchers to the field. The central objects of study of this project are modulation-invariant singular integrals and their behavior at or near the boundary of their known boundedness range. The model question, involving Carleson's maximal partial Fourier sum operator, is the characterization of the sharp integrability order sufficient for the almost-everywhere pointwise convergence of the Fourier series of a periodic function. The second, deeply related question concerns the extension of the Lacey-Thiele Holder-type estimates for the bilinear Hilbert transform to the boundary of the known range. Together with his collaborators, the principal investigator has recently obtained the current best results for both problems, relying in particular on a newly developed Calderon-Zygmund decomposition adapted to the modulation-invariant setting. It is expected that further developments of this technique will lead to additional improvements towards the solution of these two central questions, as well as of other significant open problems. A standout question is the extension of the known uniform estimates for the bilinear Hilbert transform to the full expected range of exponents, completing the original program of Calderon for the boundedness of the first commutator. Another central direction of the proposed investigation is the study of singular integral operators with rotational symmetries, a prime example of which is the Hilbert transform along a smooth vector field in the plane, by means of multiparameter time-frequency analysis techniques. Further improvements of the aforementioned techniques are also expected to impact on several questions concerning summability of multiple Fourier series.

谐波分析研究信号（函数）如何分解为基本谐波的叠加（具有明确指定的持续时间、强度和频率的信号）以及应用于这些分量的操作（滤波）如何影响重构信号。这种时频分解过程的变体在无数的现实应用中执行，例如音频或图像压缩和过滤、图像模式识别、数据同化和去噪。 该项目的主要目标之一是根据输入的相对大小和平滑度研究时频技术的理论可行性阈值。断层扫描成像中采用了类似的过程，其中通过沿穿透波（数学上描述为三维空间中的线）采样其密度来重建固体。 该项目将研究沿直线或曲线采样的数学玩具模型，其理论理解可能在改进分析图像重建方法的推导中发挥重要作用。 该项目的一个组成部分是对布朗大学谐波分析活跃研究小组内的研究生和本科生进行培训，其特别目的是吸引年轻有前途的研究人员进入该领域。该项目的中心研究对象是调制不变奇异积分及其在已知有界范围边界处或附近的行为。涉及卡尔森最大部分傅里叶和算子的模型问题是足以使周期函数的傅里叶级数几乎处处点收敛的锐可积阶的表征。第二个密切相关的问题涉及将双线性希尔伯特变换的 Lacey-Thiele Holder 型估计扩展到已知范围的边界。首席研究员最近与他的合作者一起获得了这两个问题的当前最佳结果，特别是依赖于新开发的适应调制不变设置的 Calderon-Zygmund 分解。预计该技术的进一步发展将带来对这两个核心问题以及其他重大开放问题的解决的进一步改进。一个突出的问题是将双线性希尔伯特变换的已知统一估计扩展到指数的完整预期范围，从而完成卡尔德隆关于第一换向器有界性的原始程序。所提出的研究的另一个中心方向是研究具有旋转对称性的奇异积分算子，其中一个主要的例子是通过多参数时频分析技术沿平面中的平滑矢量场进行希尔伯特变换。上述技术的进一步改进也有望影响有关多个傅立叶级数可求和性的几个问题。