Applications of Harmonic Analysis to Riesz Transforms and Commutators beyond the Classical Settings

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The mathematical discipline of analysis has been fundamental in understanding physical phenomena in the natural sciences and engineering. The behavior of a function (size, smoothness, quantitative information) that are solutions to differential equations are important. In understanding a function it is frequently useful to have "simpler" building blocks to work with. Particular mathematical tools that have proved extremely useful in addressing questions of these types and providing a framework to analyze these simple building blocks lie within the realm of harmonic analysis. A main goal of this project is to provide a deeper understanding of some of the simple building blocks that arise in important function spaces connected to function theory and partial differential equations by using and advancing the tools of harmonic analysis.This project outlines a research program combining recent results with motivation from function theory and operator theory to study questions related to the boundedness of commutators associated to Riesz transforms arising from differential operators and understanding the boundedness of the Riesz transform on a manifold with ends. The research direction couples the past work by the principal investigator with questions about boundedness of commutators with Riesz transforms associated to differential operators. In particular, the problems discussed are aimed at obtaining a better understanding of the differential operators and geometry where one can characterize appropriate BMO spaces in terms of commutators with Riesz transforms; equivalently demonstrate that the appropriate Hardy space possesses a weak factorization. A second research direction provides a holomorphic functional calculus on a manifold with ends and studies the open question of obtaining the boundedness of the Riesz transform using ideas from non-homogeneous harmonic analysis and the techniques developed by the principal investigator. Graduate students with whom the principal investigator works will be included in these and related projects, and will receive advising and career mentoring.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.