Fourier Analysis and Multilinear Operators

多线性算子的研究在很大程度上是由于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。研究者将利用最近取得的成功和进展，并面对完成该学科发展所需的问题。同时，将介绍和探索新的问题和创新的研究方法。奇异积分在椭圆正则性、广义Cauchy-Riemann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发挥了至关重要的作用。多线性奇异积分代表了通过谱分辨率对函数进行计算操作的成功企业的下一步，并通过傅里叶变换的揭示透镜来理解它们。在规格&amp；#64257；该提案的目标是研究具有最小正则性假设的双线性或多线性算子；发展更精确的外推和加权估计技术；对频域奇异双线性乘法器的分析目前还没有得到很好的理解，但这对该领域的进一步发展至关重要。最近对多线性算子的研究已经在谐波分析之外发现了可预见的和不可预料的应用。本提案中研究的一些问题可能对其他数学学科产生潜在影响，特别是在偏微分方程中。傅里叶分析方法提供了通过将信息分解成简单的构建块或波状分量来分析信息的方法。利用以不同频率振荡的波彼此之间的相互作用不大这一事实，在这些组件中进行特殊操作很容易。这是信号分析及其变换的基本思想，新傅立叶分析技术的发展通常转化为信息压缩、模式识别和其他科学与工程图像处理应用的方案和算法。研究者将继续与研究生和同事在其职业生涯的早期阶段进行互动。他的研究将与他的教学、培训和指导活动相结合，包括指导在他指导下工作的博士生。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。本提案中的研究将通过该学科的专业会议和说明性讲座向更广泛的受众传播，旨在提高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.