Endpoint Behavior of Modulation Invariant Singular Integrals

谐波分析研究信号(功能)如何分解成基波的叠加--具有明确规定的持续时间、强度和频率的信号--以及对这些分量应用的运算(滤波)如何影响重构信号。这种时频分解过程的变体在无数真实世界的应用中执行，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。该项目的广泛目标之一是根据输入的相对大小和平稳性来研究时频技术的理论可行性阈值。在层析成像中采用了一种类似的程序，即通过沿穿透波(在数学上描述为三维空间中的线)采样其密度来重建固体。本项目将研究沿直线或曲线采样的数学玩具模型，其理论理解可能对改进的解析图像重建方法的推导起到重要作用。该项目的一个组成部分是对布朗大学调和分析活跃研究小组中的研究生和本科生进行培训，目的是吸引年轻和有前途的研究人员进入该领域。这个项目的中心研究对象是调制不变的奇异积分及其在已知有界性范围的边界或附近的行为。涉及Carleson极大部分傅立叶和算子的模型问题是周期函数的傅里叶级数几乎处处逐点收敛的尖锐可积阶的刻画。第二个深相关的问题是关于双线性Hilbert变换的Lacey-Thiele Holder型估计在已知范围的边界上的推广。这位首席研究员和他的合作者最近针对这两个问题获得了目前最好的结果，特别是依赖于新开发的适应于调制不变设置的Calderon-Zygmund分解。预计这项技术的进一步发展将导致进一步改进这两个中心问题以及其他重大未决问题的解决。一个突出的问题是将已知的双线性Hilbert变换的一致估计推广到指数的全部期望范围，从而完成了Calderon关于第一个交换子有界性的原始程序。研究的另一个中心方向是利用多参数时频分析技术研究具有旋转对称性的奇异积分算子，其中最典型的例子是沿平面光滑向量场的Hilbert变换。前述技术的进一步改进也有望影响与多重傅立叶级数的可加性有关的几个问题。

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.