Averages and maximal averages over Product j-varieties in finite fields

2021/05/07 by Doowon Koh, Sujin Lee, Koh, Doowon +1
Mathematics · #Advanced Harmonic Analysis Research #Nonlinear Partial Differential Equations #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2105.03520

Abstract

We study both averaging and maximal averaging problems for Product j-varieties defined by Π_j=\x∈ \mathbb F_q^d: ∏_k=1^d x_k=j\ for j∈ \mathbb F_q^*, where \mathbb F_q^d denotes a d-dimensional vector space over the finite field \mathbb F_q with q elements. We prove the sharp L^p→ L^r boundedness of averaging operators associated to Product j-varieties. We also obtain the optimal L^p estimate for a maximal averaging operator related to a family of Product j-varieties \Π_j\_{j∈ \mathbb Fq^*}.

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