Topics in Harmonic Analysis: Interplay between Time-Frequency Analysis, Additive Combinatorics and Partial Differential Equations

Purdue University, West Lafayette IN

Investigators

Abstract

This research project focuses on the interplay between three major areas in mathematics: harmonic analysis, additive combinatorics, and partial differential equations (PDE), with the harmonic analysis component playing the dominant role. The research plan has the following characteristics: (1) it is structured into several classes of problems (some to be detailed in the next paragraph), on some of which the principal investigator has already made relevant contribution/advancements by developing new analysis methods; (2) it covers a wide range of techniques, with levels of difficulty varying depending on the nature of the subject that also allow partial progress; (3) it has two main components, namely, a purely theoretical one dealing with problems that focus on understanding the behavior of various so-called maximal operators with highly oscillatory kernels and a more applied side, part of the area of fluid dynamics, studying the problem of singularity formation for two-fluid interfaces. An advancement in this latter direction will contribute to a better understanding of the physical reality around us. Further aspects of the project are the dissemination of the principal investigator's work through conference participation, lectures, and graduate courses, interaction with other researchers/experts in the areas of harmonic analysis and PDE, the training of graduate students and postdoctoral fellows. This project investigates several problems that position themselves at the interface either between harmonic analysis and additive combinatorics, or between harmonic analysis and fluid dynamics. It is structured along several major themes, among which we mention the following (items (2), (3), and (4) represent joint projects with collaborators): (1) pointwise convergence of Fourier series near L^1; (2) the maximal Schrodinger operator, Kakeya extremizers, and sum-product estimates; (3) formation of singularities for the two-dimensional, two-fluid interface in the absence of vorticity in order to increase understanding of the "limiting" water-wave case; (4) the rotational two-dimensional water-wave problem and quasilinear systems of Klein--Gordon equations in three dimensions with nontrivial vorticity, with a focus on the time of existence for small initial data. On the first topic the author has already made contributions, developing new methods that either improved on known results (e.g., the pointwise convergence along lacunary subsequences of partial Fourier sums) or recovered the best known results stemming from the celebrated work of Lennart Carleson. The approaches for both the first and second topics combine techniques from time-frequency analysis and additive combinatorics. The third theme investigates, for the two-dimensional, two-fluid problem, how the geometry of the interface changes as the boundary approaches a "splash" scenario from the water-wave case; the fourth one explores how the time of existence for solutions (relative to small initial data) of the two-dimensional water-wave case and three dimensional Klein-Gordon systems are affected by the presence of vorticity, a new and hitherto mostly unexplored theme in the fluid dynamics literature.