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1 results

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

  • Doowon Koh, Chun-Yen Shen
  • Journal:
    Canadian Journal of Mathematics / Volume 64 / Issue 5 / 01 October 2012
    Published online by Cambridge University Press:
    20 November 2018, pp. 1036-1057
    Print publication:
    01 October 2012
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  • 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.