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Doowon Koh

  1. The generalized Erdos-Falconer distance problems in vector spaces over finite fields.

    Mon, 04/26/2010 - 15:01 — SomNambul
    Authors: Doowon Koh, Chun-Yen Shen
    Subjects: Classical Analysis and ODEs
    Abstract

    In this paper we study the generalized Erdos-Falconer distance problems in
    the finite field setting. The generalized distances are defined in terms of
    polynomials, and various formulas for sizes of distance sets are obtained. In
    particular, we develop a simple formula for estimating the cardinality of
    distance sets determined by diagonal polynomials. As a result, we generalize
    the spherical distance problems due to Iosevich and Rudnev and the cubic
    distance problems due to Iosevich and Koh. Moreover, our results are of higher
    dimensional version for Vu's work on two dimension.

  2. Extension and averaging operators for finite fields.

    Tue, 08/25/2009 - 22:41 — SomNambul
    Authors: Doowon Koh
    Subjects: Classical Analysis and ODEs
    Abstract

    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.

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