Endpoint Behavior of Modulation Invariant Singular Integrals

谐波分析研究信号（函数）如何分解成基本谐波的叠加——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实际应用中执行，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。这个项目的一个广泛目标是调查时频技术在输入的相对大小和平滑方面的理论可行性阈值。在层析成像中采用类似的程序，其中通过沿穿透波采样其密度来重建实体，数学上描述为三维空间中的线。本项目将研究沿直线或曲线采样的数学玩具模型，其理论理解可能对推导改进的分析图像重建方法起重要作用。该项目的一个组成部分是在布朗大学活跃的谐波分析研究小组中培训研究生和本科生，特别旨在吸引年轻和有前途的研究人员进入该领域。本课题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或边界附近的行为。模型问题，涉及Carleson的极大偏傅立叶和算子，是对周期函数的傅立叶级数几乎处处点向收敛的尖锐可积阶的描述。第二个与此密切相关的问题是将双线性Hilbert变换的Lacey-Thiele holder型估计扩展到已知范围的边界。与他的合作者一起，首席研究员最近获得了目前这两个问题的最佳结果，特别是依靠新开发的适用于调制不变设置的Calderon-Zygmund分解。预计这项技术的进一步发展将导致进一步改进，以解决这两个中心问题以及其他重大的未决问题。一个突出的问题是将已知的双线性希尔伯特变换的一致估计推广到指数的整个期望范围，完成了卡尔德隆关于第一对易子有界性的原始规划。本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利用多参数时频分析技术在平面上沿光滑向量场的希尔伯特变换。上述技术的进一步改进也有望对多个傅立叶级数的可和性问题产生影响。

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.