This proposal is devoted to investigations in the analysis of operators associated to singular integrals, product-like operations, function spaces and approximations, estimates for commutators and null forms, and other related problems. The main tools to be used are decomposition techniques related to time-frequency expansions. Singular multilinear pseudodifferential operators will be studied together with their application to nonlinear analysis, estimates in Sobolev spaces, and partial differential equations. Another part of the project is concerned with the mathematical modeling of transformations of quasi-ordered geometries as measured by their Fourier spectra. This is motivated by ongoing research in biology in the analysis of the scattering of light by nanostructures in the tissues of living organisms.
A common task in both theoretical and applied problems is to find the best way to decompose a complicated system in a way that efficiently quantifies observable properties and effects and keeps the available information in an organized and manageable form. Fourier analysis decodes such information by resolving a signal or function into a spectrum of waves of different amplitudes and frequencies, while providing also a way to study spatial order, structured geometries, and patterns in data. Combined with this analysis, multilinear approaches to complicated phenomena in nonlinear science will provide further understanding of problems where simple linear or first-order approximations are not sufficient. The analysis of ordered structures is also of importance to research in other areas of biology and physical sciences. For example, it is a relevant aspect in questions about the study of macroscopic physical properties of materials in terms of their microscopic structures.
