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.
The aim of the proposed research focused on the development of mathematical theories in the area of Fourier Analysis. In a broad sense, this branch of mathematics allows the study of signals, such as sounds and images, by breaking them down into fundamental pieces that are less complex and, therefore, easier to examine. Even though the problems studied are on the abstract and theoretical side, their solutions have potential applications to the fields of image and signal processing as well as the theory of nonlinear partial differential equations from areas of physics such as fluid dynamics, quantum mechanics and optics. Eight original research articles were published in peer-reviewed journals as a result of the investigations associated to this project. These publications provide new contributions on topics related to bilinear Sobolev-Poincaré type inequalities and Leibniz-type rules, boundedness properties of bilinear pseudodifferential operators, and regularity properties of solutions to degenerate p-laplacian equations. The PI has disseminated her work through websites, participation at national and international conferences, and invited talks and courses in the US and abroad. The proposed project has integrated research and education at the undergraduate, graduate and postdoctoral levels. The development of human resources has been strongly emphasized as the PI supervised research projects for three undergraduate students, one PhD student and one postdoctoral fellow. The PI has been involved in the organization of conferences at various levels including an NSF/CBMS conference. She has also extensively contributed to the curriculum development of the Mathematics Department at Kansas State University.
