Abstract
In this paper we study the mapping properties of the averaging operator over a variety given by a system of homogeneous equations over a finite field. We obtain optimal results on the averaging problems over two-dimensional varieties whose elements are common solutions of diagonal homogeneous equations. The proof is based on a careful study of algebraic and geometric properties of such varieties. In particular, we show that they are not contained in any hyperplane and are complete intersections. We also address partial results on averaging problems over arbitrary dimensional homogeneous varieties which are smooth away from the origin.
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Acknowledgments
The authors are grateful to Anthony Flatters and Tom Ward for useful discussions and in particular for the idea of the proof of Lemma 2.2. Doowon Koh was supported by the Research Grant of Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2012R1A1A1001510). Chun-Yen Shen was supported by the NSC, through Grant NSC102-2115-M-008-015-MY2. Igor Shparlinski was supported by the Research Grant of the Australian Research Council (DP130100237).
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Koh, D., Shen, CY. & Shparlinski, I. Averaging Operators Over Homogeneous Varieties Over Finite Fields. J Geom Anal 26, 1415–1441 (2016). https://doi.org/10.1007/s12220-015-9595-5
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DOI: https://doi.org/10.1007/s12220-015-9595-5