Problems Related to Fourier Restriction Estimates

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This project revolves around several fundamental questions in harmonic analysis, which is a field stemming from the study of Fourier series and Fourier transform and is closely connected with partial differential equations, number theory, geometric measure theory, and real life applications such as signal processing and compressed sensing. The Fourier transform decomposes a function of time into different frequency components, similarly to how a music chord can be expressed as the pitches of its constituent notes. Harmonic analysis studies how the time information and the frequency information interact with each other. A fundamental question (i.e. Fourier restriction problem) that has been studied for decades is how the geometry of the frequency support of the Fourier transform dictates the size of the original function in time. Various quantifications of such relations are referred to as Fourier restriction estimates and turn out to be extremely challenging to study. Even for the simplest geometric objects such as spheres and paraboloids, many questions are still wide open. Restriction estimates are interesting also due to their connection with many other problems, within or outside analysis. It is well known that restriction estimates can be applied to study the Kakeya conjecture (on the minimum area of a set containing a unit line segment in each direction), the existence and growth of solutions to Schrödinger and wave equations, and the number of solutions to Diophantine equations in number theory.This project studies several problems related to Fourier restriction estimates. First, the principal investigator intends to further the investigation of the Fourier restriction conjecture for the paraboloid and the cone via the polynomial method. This method explores the algebraic structure of the time-frequency decomposition of the function, and has shown to be extremely powerful in obtaining many state-of-the-art results in the theory. Second, the principal investigator proposes to continue the study of weighted restriction estimates (i.e. when the Lebesgue measure is replaced with a fractal measure) and apply them to estimate divergence sets of the Schrödinger or wave equations and to distance set problems (on how the size of a set dictates the size of its distance set). The major difficulty in this direction is the lack of a crucial tool (orthogonality) caused by the presence of the fractal measure. Here an interesting approach will attack the distance set problems directly by studying the behavior of the fractal measure. Last, the principal investigator wants to explore the role in restriction theory of a recently developed tool called sparse domination. This method, arising from singular integral theory, is a way to reduce the study of the original continuous, spread out operator to that of a class of much simpler dyadic, positive, local operators. This method has shown to be extremely useful in singular integral theory and has become a modern view of point in describing operators. The principal investigator plans to further the study of sparse bounds of operators related to restriction estimates and with a Kakeya nature, such as the Bochner-Riesz multipliers, singular integrals along manifolds, directional maximal operators, and multi-parameter singular integral operators.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.