arXiv 0908.32662009-08math.CA

有限域的扩张算子和平均算子

Extension and averaging operators for finite fields

Doowon Koh, Chun-Yen Shen

备注：: 14 pages, published version

摘要：

本文研究了与代数簇$S=\{x\in {\mathbb F}_q^d: Q(x)=0\}$相关的扩张算子和平均算子的$L^p-L^r$估计，其中$Q(x)$是有限域${\mathbb F}_q.$上的非退化二次型.当$d\geq 3$是奇数且曲面$S$包含一个$(d-1)/2$维子空间时，我们得到了指数$r$，其中$L^2-L^r$扩张估计是尖锐的.特别地，我们给出了与三维特定表面$S$相关的延拓问题的完整解。在偶数维$d\geq 2$中，我们还研究了尖锐的$L^2-L^r$扩展估计。这样的结果是广义的版本和扩展到更高的维度的圆锥扩张问题，Mochenhaupt和陶研究在三维空间。研究了平均算子在曲面$S$上的有界性。在奇数维$d\geq 3$，我们完全解决的问题，为$L^p-L^r$估计的平均运营商有关的表面$S.$另一方面，在这种情况下，当$d\geq 2$是偶数和$S$包含一个$d/2$维的子空间，使用我们的最佳$L^2-L^r$结果的扩展定理，我们，除了端点，有尖锐的$L^p-L^r$估计的平均运营商的表面$S$在偶数维。

英文摘要：

In this paper we study $L^p-L^r$ estimates of both extension operators and averaging operators associated with the algebraic variety $S=\{x\in {\mathbb F}_q^d: Q(x)=0\}$ where $Q(x)$ is a nondegenerate quadratic form over the finite field ${\mathbb F}_q.$ In the case when $d\geq 3$ is odd and the surface $S$ contains a $(d-1)/2$-dimensional subspace, we obtain the exponent $r$ where the $L^2-L^r$ extension estimate is sharp. In particular, we give the complete solution to the extension problems related to specific surfaces $S$ in three dimension. In even dimensions $d\geq 2$, we also investigates the sharp $L^2-L^r$ extension estimate. Such results are of the generalized version and extension to higher dimensions for the conical extension problems which Mochenhaupt and Tao studied in three dimensions. The boundedness of averaging operators over the surface $S$ is also studied. In odd dimensions $d\geq 3$ we completely solve the problems for $L^p-L^r$ estimates of averaging operators related to the surface $S.$ On the other hand, in the case when $d\geq 2$ is even and $S$ contains a $d/2$-dimensional subspace, using our optimal $L^2-L^r$ results for extension theorems we, except for endpoints, have the sharp $L^p-L^r$ estimates of the averaging operator over the surface $S$ in even dimensions.

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摘要：

英文摘要：

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