This mathematical project focuses on further development of modern techniques in harmonic analysis. Harmonic analysis has proven to be an invaluable tool for other branches of mathematics, with applications in physics, natural sciences, signal and digital image processing, and engineering. The project aims to study oscillatory integrals and maximal functions. The class of oscillatory integrals considered here encompasses various integral transforms such as the Fourier transform, which is a basic object in science as it allows the decomposition of a signal into its fundamental frequencies. Quantitative statements about the maximal functions associated to a family of objects often lead to qualitative conclusions about the limiting behavior of that family. In particular, maximal functions play an important role in understanding differentiability properties of functions and properties of the solutions of the partial differential equations that govern the laws of physics and nature. This project will contribute to workforce development through mentoring of undergraduate students and to infrastructure development via conference organization.
The investigator and collaborators will work on several interrelated projects in harmonic analysis that involve the study of Fourier integral operators (FIOs) and maximal functions. There are three main research directions. The first corresponds to the applications of decoupling inequalities and local smoothing estimates for FIOs towards establishing sharp Lp-Sobolev bounds of averaging operators over curves and sharp Lp bounds for related maximal functions, associated to families of such averages. Related questions such as variable coefficient analogues and averaging operators over degenerate manifolds will also be studied. The research will also investigate regularity properties of those maximal functions for functions in first order Sobolev spaces. This connects with the second theme of this project, establishing endpoint Sobolev regularity properties of the Hardy-Littlewood maximal function and its fractional counterpart. The techniques here are non-Fourier analytic. The third and final direction of research is devoted to the study of two-weighted inequalities of Fefferman-Stein type for the half-wave propagator and non-degenerate Fourier integral operators, in which the weights are related via a novel maximal function associated to the wave front set of the FIO.
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.
