Extension and averaging operators for finite fields
Journal
Proceedings of the Edinburgh Mathematical Society
Journal Volume
56
Journal Issue
2
Pages
599-614
Date Issued
2013
Author(s)
Koh, D.
Abstract
Abstract In this paper we study L p −L r 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 L 2 − L 4 conical extension result in , due to Mockenhaupt and Tao, to the L 2 − L 2( 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.
Type
journal article