Fourier Analysis Techniques, Operators, and Function Spaces

该建议侧重于傅立叶分析和奇异积分研究的各个领域的新的和具有挑战性的问题。一些问题涉及多线性算子。托雷斯将工作的问题，涉及范数估计变系数双线性奇异变换和双线性pseudodiomatic运营商，极大函数的多线性卡尔德龙-Zygmund理论，多线性权重，和多线性强奇异积分算子。特别是，几个问题的规律性的符号和内核的运营商调查，端点估计，勒贝格和Sobolev空间估计，以及相关问题将被考虑。PI还将使用有关函数样本的信息对各种空间的特征进行研究。最后，托雷斯将继续研究数学公式，量化与傅立叶分析的准有序几何。往往是一个很难的分析问题的解决方案不能实现，直到有效的方法来分解功能和转换被发现。傅立叶分析分解成一个功能的叠加波振荡的时间与特定的频率。其他相关的时间-频率分解通过在多个尺度或分辨率级别上编码信息来完善这种分析。这允许理解函数的更有趣的属性和转换它们的操作。这些技术在科学和工程的许多领域已经成为非常有价值的工具。应用于信息压缩、音频过滤、卫星摄影和其他形式的图像处理。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。预计这一领域的新进展将继续对其他学科的问题产生影响，在这些学科中，需要量化复杂的多尺度信息并以连贯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.