Restriction of averaging operators to algebraic varieties over finite fields

2016/01/28 by Doowon Koh, Koh, Doowon, Seongjun Yeom +1
Computer Science · Mathematics · #11T23 (Secondary) #42B05 (Primary) #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1601.07677

openalex publication_date 2016/01/28 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We study L^p→ 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⊂ \mathbb F_q^d. As a consequence, we obtain optimal results on the restricted averaging problems for spheres and paraboloids in dimensions d≥2, and cones in odd dimensions d≥ 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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