Virginia Naibo

This research project concerns bilinear Fourier analysis. Broadly speaking, Fourier analysis is a mathematical discipline for the study of signals, such as sound and images. The study of signals by way of Fourier analysis involves breaking them down into fundamental pieces that are less complex and, therefore, easier to examine. Information obtained from the individual pieces is then synthesized to obtain information about the original signal. Fourier analysis has had far-reaching applications in other areas of mathematics, physics, engineering, medicine, industry, and the applied sciences. This project will investigate central questions in the field of bilinear Fourier analysis, where a pair of signals are analyzed simultaneously. The outcomes are anticipated to have applications in the theory of partial differential equations, to topics as diverse as fluid dynamics, quantum mechanics, and optics. The project will also contribute to the integration of research and education at the graduate and undergraduate levels. The project aims to contribute to new developments in bilinear Fourier analysis through the investigation of a suite of interrelated questions motivated by applications to analysis and partial differential equations. The project will investigate several approaches, based on tools including representations of functions, Littlewood-Paley techniques, and symbolic calculus, to generate a host of new bilinear estimates and boundedness properties of bilinear pseudodifferential operators. The results are expected to apply to the pointwise multiplication properties of function spaces, local well-posedness results for the Euler equations and the ideal magnetohydrodynamic equations, and scattering properties of solutions of systems of partial differential equations associated to local and nonlocal operators. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

The subject matter of this project belongs to the realm of bilinear Fourier analysis. After the pioneering work of Joseph Fourier in the first decades of the nineteenth century, one is now familiar with the process of decomposing a signal or function into its elementary frequency components (analysis) as well as with the reverse process of superposing individual frequency components to form a single signal (synthesis). Fourier analysis thus acts in a way similar to a prism, which allows one to see the individual color components of a beam of light. Along these lines, when two functions or signals coexist, their frequency components interact and this phenomenon plays a key role in the study of certain partial differential equations that arise, for instance, in optics, quantum mechanics, and fluid dynamics. In the field of bilinear Fourier analysis, tools are developed to model the behavior and interaction of two signals by decomposing each one into their constituent frequencies, separating each decomposition into low and high frequencies, and studying the interplay between the low-low, high-low, and high-high frequencies from each decomposition. This project will also contribute to the integration of research and education at the postdoctoral, graduate, and undergraduate levels, to advancing discovery, to forming human resources, and to developing academic curricula. Motivated by the study of commutators, bilinear Leibniz-type rules, paraproducts, and related topics in analysis and partial differential equations, the research activities of this project aim at developing methods in bilinear Fourier analysis to advance the theory of bilinear pseudo-differential operators and their applications. In particular, problems to be addressed include the description of the mapping properties, in the scales of Lebesgue, Besov, and Triebel-Lizorkin spaces, of bilinear pseudodifferential operators with symbols in certain critical classes.

This award will support an NSF/CMBS regional conference to be held at Kansas State University in the summer of 2013 on the global behavior of solutions to critical nonlinear wave equations. The principal speaker will be Professor Carlos Kenig from the University of Chicago. The lectures will cover such areas as local and global well-posedness, scattering, finite time blow up, and soliton resolution for classes of nonlinear dispersive equations. In addition to the 10 main lectures by Dr. Kenig, lectures will be presented by other experts in these fields. Hallmarks of the NSF/CBMS regional conference series are focus on a single important and timely area of research by a leading practitioner, a published monograph for a wider audience, and continued effect and local stimulation through regional recruitment emphasis. For this conference, the lectures will also be posted online to provide greater access. This award will support approximately 25 participants in the conference, mostly at early stages of their careers.

This proposal will advance the investigator's research in the areas of Fourier and real analysis. The posed problems lie at the core of the theories of bilinear pseudodifferential operators and weighted bilinear Poincaré and Sobolev inequalities and include the development and implementation of bilinear techniques at their most fundamental level in time-frequency and real analysis, thus broadening the scope of their applications to Analysis and Partial Differential Equations. To further these ends, particular attention is given to the study of boundedness properties in the setting of Lebesgue and modulation spaces of bilinear pseudodifferential operators and molecular paraproducts, and to weighted bilinear Poincaré and Sobolev inequalities through the study of bilinear representation formulas and bilinear fractional integral operators in the context of Carnot-Carathéodory spaces. Other relevant function spaces considered in this research program include the scales of Sobolev, Besov, Triebel-Lizorkin spaces, BMO, weak Lebesgue spaces, and Campanato-Morrey spaces as well as their weighted versions. Fourier Analysis has since its origins made many significant contributions to various areas of mathematics, physics and engineering; the research developed through this program will positively continue to add to these disciplines. The proposed research has in particular applications to the theory of nonlinear Partial Differential Equations from areas of physics such as fluid dynamics, quantum mechanics and optics. This project will also contribute to the integration of research and education at the postdoctoral, graduate, and undergraduate levels, to advancing discovery, forming human resources, and developing academic curriculum.