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.
