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arXiv · 1601.07677

Restriction of averaging operators to algebraic varieties over finite fields

Doowon Koh·Seongjun Yeom
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Abstract

We study $L^p\to L^r$ estimates for restricted averaging operators related to algebraic varieties $V$ of $d$-dimensional vector spaces over finite fields $\mathbb F_q$ with $q$ elements. We observe properties of both the Fourier restriction operator and the averaging operator over $V\subset \mathbb F_q^d.$ As a consequence, we obtain optimal results on the restricted averaging problems for spheres and paraboloids in dimensions $d\ge2,$ and cones in odd dimensions $d\ge 3.$ In addition, when the variety $V$ is a cone lying in an even dimensional vector space over $\mathbb F_q$ and $-1$ is a square number in $\mathbb F_q$, we also obtain sharp estimates except for two endpoints.

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Doowon Koh, Seongjun Yeom. 2016-01-28. Restriction of averaging operators to algebraic varieties over finite fields. https://arxiv.org/abs/1601.07677

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