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Extension and averaging operators for finite fields

Part of: Finite fields and commutative rings (number-theoretic aspects) Harmonic analysis in several variables

Published online by Cambridge University Press:  30 April 2013

Doowon Koh  and
Chun-Yen Shen
Doowon Koh
Affiliation:
Department of Mathematics, Chungbuk National University, Cheongju City, Chungbuk-Do 361-763, Republic of Korea ([email protected])
Chun-Yen Shen
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton L8S 4K1, Canada ([email protected])
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Abstract

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In this paper we study Lp−Lr estimates of both the extension operator and the averaging operator associated with the algebraic variety S = {x ∈ : Q(x) = 0}, where Q(x) is a non-degenerate quadratic form over the finite field with q elements. We show that the Fourier decay estimate on S is good enough to establish the sharp averaging estimates in odd dimensions. In addition, the Fourier decay estimate enables us to simply extend the sharp L2−L4 conical extension result in , due to Mockenhaupt and Tao, to the L2−L2(d+1)/(d−1) estimate in all odd dimensions d ≥ 3. We also establish a sharp estimate of the mapping properties of the average operators in the case when the variety S in even dimensions d ≥ 4 contains a d/2-dimensional subspace.

Keywords

extension problemaveraging operatorfinite field

MSC classification

Secondary: 42B05: Fourier series and coefficients 11T24: Other character sums and Gauss sums
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 2 , June 2013 , pp. 599 - 614
Copyright
Copyright © Edinburgh Mathematical Society 2013

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