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 $d \geq 2$, and cones in odd dimensions $d \geq 3$. 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.
Citation
Doowon Koh. Seongjun Yeom. "Restriction of Averaging Operators to Algebraic Varieties over Finite Fields." Taiwanese J. Math. 21 (1) 211 - 229, 2017. https://doi.org/10.11650/tjm.21.2017.7743
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