Modern Harmonic Analysis Techniques in Complex Variables and Non-Homogeneous Settings

Harmonic analysis is a major branch of mathematical analysis which focuses on studying the behavior of functions by breaking them down into simpler and easier-to-understand component parts. The developments in harmonic analysis have led to concrete advances in medical imaging, image compression algorithms, signal processing, and neuroscience. This project examines questions in harmonic analysis and related fields from a more theoretical or pure perspective of basic research. As part of this award, the PI also mentors undergraduate students in research projects, which increases the STEM pipeline and supports higher education and society at large. This project consists of two main streams: harmonic analysis in the special setting of non-doubling measures and applying harmonic analysis to problems in complex analysis which also connect to operator theory. In the context of non-homogeneous harmonic analysis, questions relating to a novel paradigm for the sparse domination of Calderon-Zygmund operators and commutators are considered. Within the second stream, the PI investigates two-weight and endpoint commutator estimates for the Bergman projection, Lp estimates for the Cauchy-Szego and Bergman projections on Lipschitz and other minimally smooth domains, and two-weight inequalities for the Bergman projection on the unit disk. The specific tools used to study these questions include a non-homogeneous Calderon-Zygmund decomposition, an approach to the study of holomorphic projection operators originated by Kerzman and Stein, and using weighted Haar decompositions and random dyadic grids to achieve two-weight estimates. 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.