Conjecture and improved extension theorems for paraboloids in the finite\n field setting

2016/03/21 by Doowon Koh, Koh, Doowon
Engineering · Computer Science · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Advanced Mathematical Modeling in Engineering #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.1603.06512

Abstract

We study the extension estimates for paraboloids in d-dimensional vector\nspaces over finite fields F_q with q elements. We use the connection between\nL² based restriction estimates and L^p\→ L^r extension estimates for\nparaboloids. As a consequence, we improve the L²\→ L^r extension results\nobtained by A. Lewko and M. Lewko in even dimensions d\≥ 6 and odd dimensions\nd=4\ℓ+3 for \ℓ \∈ mathbb N. Our results extend the consequences for 3-D\nparaboloids due to M. Lewko to higher dimensions. We also clarifies conjectures\non finite field extension problems for paraboloids.\n

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