Bilinear techniques in time-frequency and real analysis

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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.