Fourier Analysis and Multilinear Operators

多线性算子的研究在很大程度上是通过对非常具体和自然出现的例子的兴趣而发展起来的，这些例子的理解产生了强大的时频工具。调查员将利用最近取得的成功和进展，面对完成这一主题发展所需的问题。同时，还将介绍和探索研究这些问题的新颖问题和创新方法。奇异积分在椭圆正则性、广义柯西-黎曼方程、Sobolev嵌入、Littlewood-Paley理论以及偏微分方程和函数空间研究中的其他问题中起着至关重要的作用。多线性奇异积分代表了通过其频谱分辨率和通过揭示傅里叶变换的透镜对函数的理解来对函数进行计算运算的已被证明成功的企业的下一步。该提议的特殊目标包括研究具有最小正则性假设的双线性或多线性算子；开发更精确的外推和加权估计技术；以及分析在频域中具有奇异性的双线性乘子，但这对该领域的进一步发展至关重要。最近关于多线性算子的工作已经在调和分析之外发现了可预见的和意想不到的应用。这项建议中要研究的一些问题可能会对其他数学学科产生潜在影响，特别是在偏微分方程式方面。傅立叶分析方法提供了通过将信息分解成简单的积木或波状分量来分析信息的方法。通过利用以不同频率振荡的波彼此之间不太相互作用的事实，在这些组件中很容易执行特定的操作。这是分析信号及其变换的基本思想，新傅里叶分析技术的发展进展通常转化为用于信息压缩、模式识别和其他图像处理在科学和工程中的应用的方案和算法。调查员将继续与处于职业生涯早期阶段的研究生和同事互动。他的研究将与他的教学、培训和指导活动相结合，其中包括在他的指导下指导博士生工作。他还将继续为本科生提供研究机会，并作为教师导师参与现有项目，以增加他所在机构的学生多样性。这项建议中的研究将通过该学科的专业会议以及面向更广泛受众的说明性讲座来传播，其目的是提高普通公众对数学和科学的认识。

The study of multilinear operators has developed largely by the interest in very concrete and naturally appearing examples whose understanding has produced powerful time-frequency tools. The investigator will capitalize on the recent successes and progresses achieved and confront problems that are needed to complete the development of the subject. At the same time novel problems and innovative approaches to study them will be introduced and explored. Singular integrals have played a crucial role in questions related to elliptic regularity, generalized Cauchy-Riemann equations, Sobolev embeddings, Littlewood-Paley theory, and other problems in partial differential equations and the study of function spaces. Multilinear singular integrals represent a next step in the proven successful enterprise of computing operations on functions via their spectral resolution and the understanding of them through the revealing lenses of the Fourier transform. Among the specific goals of the proposal are the study of bilinear or multilinear operators with minimal regularity assumptions; the development of more precise extrapolation and weighted estimates techniques; and the analysis of bilinear multipliers with singularities in the frequency domain not well-understood yet, but which are of critical importance for further progress in the field. Recent work on multilinear operators has already found both foreseen as well as unexpected applications outside harmonic analysis. Some of the questions to be investigated in this proposal can have potential impact in other mathematical disciplines, in particular in partial differential equations. Fourier analysis methods provide ways to analyze information by decomposing it into simple building blocks or wavelike components. Particular operations are easy to perform in these components by exploiting the fact that waves which oscillate at different frequencies do not interact much with each other. This is a basic idea in the analysis of signals and their transformations, and progress in the development of new Fourier analysis techniques often translates into schemes and algorithms for compression of information, pattern recognition, and other application of image processing in science and engineering. The investigator will continue to interact with graduate students and colleagues in the early stages of their careers. His research will be integrated with his teaching, training, and mentoring activities, which include the direction of PhD students working under his supervision. He will also continue to conduct research opportunities for undergraduates and participate as faculty mentor in existing programs to increase student diversity at his institution. The research in this proposal will be disseminated both through professional conferences in the discipline as well as expository lectures to broader audiences, which are intended to increase the awareness about mathematics and science in the general public.