Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields

Koh, Doowon ; Shen, Chun-Yen

Canadian journal of mathematics, Tome 64 (2012) no. 5, pp. 1036-1057

Résumé

In this paper we study the extension problem, the averaging problem, and the generalized Erdős–Falconer distance problem associated with arbitrary homogeneous varieties in three dimensional vector spaces over finite fields. In the case when the varieties do not contain any plane passing through the origin, we obtain the best possible results on the aforementioned three problems. In particular, our result on the extension problem modestly generalizes the result by Mockenhaupt and Tao who studied the particular conical extension problem. In addition, investigating the Fourier decay on homogeneous varieties enables us to give complete mapping properties of averaging operators. Moreover, we improve the size condition on a set such that the cardinality of its distance set is nontrivial.

Détail

DOI : 10.4153/CJM-2011-089-x

Mots-clés : 42B05, 11T24, 52C17, extension problems, averaging operator, finite fields, Erdős–Falconer distance problems, homogeneous polynomials

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