Fourier Analysis Techniques, Operators, and Function Spaces

这项建议集中在傅立叶分析和奇异积分研究的各个领域中的新的和具有挑战性的问题上。其中几个问题涉及到多线性算子。Torres将致力于解决变系数双线性奇异变换和双线性伪微分算子的范数估计、多线性Calderón-Zygmund理论的极大函数、多线性权重和多线性强奇异积分算子的问题。特别是，关于所研究的算子的符号和核的正则性的几个问题，端点估计，勒贝格和索博列夫空间估计，以及相关问题将被考虑。PI还将利用函数样本的信息对各种空间的刻画进行研究。最后，托雷斯将继续研究与傅立叶分析有关的量化准有序几何的数学公式。通常，只有找到有效的方法来分解函数和变换，才能解决分析中的难题。傅立叶分析将函数分解为在时间上以特定频率振荡的波的叠加。其他相关的时频分解通过在多个尺度或分辨率级别上对信息进行编码来改进这种分析。这使我们能够理解更有趣的函数属性以及转换它们的操作。这些技术已经在科学和工程的许多领域成为非常有价值的工具。在信息压缩、音频过滤、卫星摄影和其他形式的图像处理中都有应用。通过最初在数学分析中所做的研究，时频技术在应用中的极端有效性已经变得显而易见。这一领域的新进展预计将继续对其他学科的问题产生影响，这些学科需要量化复杂的多尺度信息，并以协调一致的方式组织这些信息。

This proposal focuses on new and challenging problems in various areas of Fourier analysis and the study of singular integrals. Several of the problems involve mutilinear operators. Torres will work on problems involving norm estimates for variable coefficient bilinear singular transforms and bilinear pseudodifferential operators, maximal functions for the multilinear Calderón-Zygmund theory, multilinear weights, and multilinear strongly singular integral operators. In particular, several issues about the regularity of the symbols and kernels of the operators investigated, end-point estimates, Lebesgue and Sobolev space estimates, and related problems will be considered. The PI will also carry on research on the characterization of various spaces using information about the samples of functions. Finally, Torres will continue to investigate mathematical formulations that quantify quasi-ordered geometries in connection with Fourier analysis.Quite often the solution of a hard problem in analysis cannot be achieved until effective ways to decompose functions and transformations are found. Fourier analysis decomposes a function into a superposition of waves oscillating in time with specific frequencies. Other related time-frequency decompositions refine this analysis by codifying information at multiple scales or levels of resolution. This permits the understanding of more interesting properties of functions and the operations that transform them. The techniques have established themselves as very valuable tools in many areas of science and engineering. Applications are found in compression of information, audio filtering, satellite photography, and other forms of image processing. The extreme effectiveness in applications of the time-frequency techniques has become evident through research originally done in mathematical analysis. New progress in this area is expected to continue to have impact in problems in other disciplines where there is a need to quantify complicated multi-scale information and organize it in a coherent way.