Document Type
Honors Project - Open Access
Abstract
The restriction conjecture asks for a meaningful restriction of the Fourier transform of a function to a sufficiently curved lower dimensional manifold. It then conjectures certain size estimates for this restriction in terms of the size of the original function. It has been proven in 2 dimensions, but it is open in dimensions 3 and larger, and is an area of much recent active effort. In our study, instead of aiming to prove the restriction conjecture, we target understanding its worst-case scenarios within known estimates. Specifically, we investigate the extension operator applied to antipodally concentrating profiles, examining the ratio of the norms of these extensions. This involves understanding how the mass near the north pole compares to the mass near the south pole in terms of magnitude. Initial computational studies confirmed the established dichotomy between p>2 and 1≤p≤2. Based on these findings, we propose two conjectures: the first one is that there are 3 cases of the behavior of this constant, and the second one is that there exists a cutoff. We will also present some facts and conjectures related to special values such as the endpoint of t=1.
Recommended Citation
Liu, Hongyi, "A Discussion on Estimation of the Best Constant for Spherical Restriction Inequalities" (2024). Mathematics, Statistics, and Computer Science Honors Projects. 89.
https://digitalcommons.macalester.edu/mathcs_honors/89
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