Restriction Estimates and General Oscillatory Integrals

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This research project concerns work in harmonic analysis, also known as Fourier analysis. Harmonic analysis is a subject about the Fourier transform, which decomposes a function into its constituent frequencies. In other words, by taking the Fourier transform we effectively write a function as a superposition of monochromatic waves (waves with only one frequency). Of particular interest are the following restriction type problems: How large can a function be, in the pointwise sense or in some averaged sense, if its frequency lives in a particular restricted set? People have found that if the restricted frequency set is "curved", for example being the unit sphere, very nontrivial estimates about the function can be obtained/expected. This is useful in understanding physics phenomena dictated by certain natural partial differential equations such as the Schrodinger equation or the wave equation. Moreover, restriction type problems also turn out to be the key to the understanding of certain behaviors of natural numbers. It turns out that, for example, once people understand well on properties on waves whose frequencies are perfect 10-th powers, they can consequently answer a variety of questions on representing a natural number as a sum of a few perfect 10-th powers.In this project, the PI proposes to study restriction type problems: If we have a subset M (usually a curved submanifold or a fractal set) in the frequency space and some measure in the physical space, we want to bound some norm of a function with respect of the given measure whenever the frequency support of that function is in the given M. One proposed direction is Stein's Restriction Conjecture, where the measure is just the Lebesgue measure and the manifold is the unit paraboloid. The PI is also interested in the situations when the measure is a fractal measure, or when M is a moment manifold or a fractal set. For most questions of this type, the optimal estimates are far from being well understood. The goal of this research would be to obtain new estimates (i.e. estimates with improved exponents) in the above setting. In particular, it is anticipated that improvements on Stein's conjecture can be obtained when this project develops. For the proposed approach, the PI anticipates a subset of analytic (induction on scales, decoupling and refined Strichartz type reasoning), algebraic (the polynomial method, differential geometry and real algebraic geometry), combinatorial (Multilinear Kakeya, sum-product theory, etc.) and geometric measure theoretic (radial projection theory, etc.) tools can come into play. He also anticipates emerging new connections between harmonic analysis and nearby areas will arise.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.