Weights in Harmonic Analysis and PDEs

University Of Alabama Tuscaloosa, Tuscaloosa AL

Investigators

Abstract

This project concerns two areas within the field of mathematical analysis, namely harmonic analysis and partial differential equations. Both have proved to be very effective in understanding a variety of physical phenomena and have wide applications in engineering and the natural sciences. Partial differential equations are a natural way to model dynamic processes (that is, processes that evolve or change in some way). Harmonic analysis provides both a firm theoretical foundation on which to construct these models and effective tools for analyzing their behavior. One of the main goals of this research is to expand our knowledge of harmonic analysis and its applications to the study of partial differential equations. Significant parts of this project include education and mentoring of graduate students, particularly women and under-represented minorities, and the development of new international research collaborations. The principal investigator (PI) is working on two projects in harmonic analysis and partial differential equations. In the first, the PI is studying matrix weighted estimates for singular and fractional integrals. He is proving generalizations of the Rubio de Francia extrapolation theorem in this setting and developing a theory of matrix weighted Hardy spaces and matrix weighted variable Lebesgue spaces. These results generalize the extensive literature on scalar weighted inequalities and highlight the differences between scalar and matrix weights. New techniques involving convex-set valued functions are used to overcome various technical obstacles that arise in the passage from scalar to matrix weights. In the second project, the PI is studying the existence, uniqueness, and regularity properties of solutions of second order, degenerate elliptic equations with lower order terms. The goal is to construct a theory on as general an equation as possible with the fewest assumptions on the coefficients and the region. These assumptions are expressed in terms of the existence of matrix weighted Sobolev and Poincare ́ inequalities. This approach unites and extends a number of results that are already in the literature. The PI is also studying the existence of such Sobolev and Poincare ́ inequalities by applying the theory of matrix weighted norm inequalities. 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.