Methods and Applications for Bilinear Operators

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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.