Endpoint Behavior of Modulation Invariant Singular Integrals

谐波分析研究信号（函数）如何分解为基本谐波的叠加-具有明确规定的持续时间，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实际应用中被执行，例如音频或图像压缩和滤波、图像模式识别、数据同化和去噪。 该项目的主要目标之一是研究时频技术在输入的相对大小和平滑度方面的理论可行性阈值。在断层摄影成像中采用类似的过程，其中通过沿着穿透波沿着对其密度进行采样来重建固体，穿透波在数学上被描述为三维空间中的线。 本计画将研究沿着直线或曲线取样的数学玩具模型，其理论上的了解可能在衍生改良的解析影像重建方法上扮演重要的角色。 该项目的一个组成部分是在布朗大学谐波分析积极研究小组内培训研究生和本科生，特别是为了吸引年轻和有前途的研究人员进入该领域。该项目的主要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。模型的问题，涉及Carleson的最大部分傅立叶和算子，是充分的几乎处处逐点收敛的傅立叶级数的周期函数的尖锐的可积阶的表征。第二，深相关的问题涉及扩展的Lacey-Thiele霍尔德型估计的双线性希尔伯特变换的边界的已知范围。与他的合作者一起，主要研究人员最近获得了目前最好的结果，这两个问题，特别是依赖于新开发的卡尔德龙-齐格蒙德分解适应调制不变的设置。预计这种技术的进一步发展将导致对这两个中心问题的解决方案，以及其他重要的开放问题的进一步改进。一个突出的问题是扩展的双线性希尔伯特变换的指数的全预期范围内的已知的统一估计，完成原来的计划卡尔德龙的有界性的第一个换向器。建议调查的另一个中心方向是研究奇异积分算子的旋转对称性，其中一个主要的例子是希尔伯特变换沿着一个光滑的向量场在平面上，通过多参数时频分析技术。上述技术的进一步改进，预计也将影响到几个问题的多个傅立叶级数的求和。

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.