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Joshua Zahl

  1. A Szemeredi-Trotter type theorem in $\mathbb{R}^4$.

    Чт, 03/22/2012 - 15:01 — SomNambul
    Authors: Joshua Zahl
    Subjects: Combinatorics
    Abstract

    We show that under suitable non-degeneracy conditions, $m$ points and $n$
    2--dimensional algebraic surfaces in $\mathbb{R}^4$ satisfying certain
    "pseudoflat" requirements can have at most $O(m^{2/3}n^{2/3} + m + n)$
    incidences, provided that $m\leq n^{2-\epsilon}$ for any $\epsilon>0$ (where
    the implicit constant in the above bound depends on $\epsilon$), or $m\geq
    n^2$. As a special case, we obtain the Szemer\'edi-Trotter theorem for
    2--planes in $\mathbb{R}^4,$ again provided $m\leq n^{2-\epsilon}$ or $m\geq
    n^2$.

  2. L^3 estimates for an algebraic variable coefficient Wolff circular maximal function.

    Пнд, 12/06/2010 - 16:01 — SomNambul
    Authors: Joshua Zahl
    Subjects: Classical Analysis and ODEs
    Abstract

    In 1997, Thomas Wolff proved sharp $L^3$ bounds for his circular maximal
    function, and in 1999, Kolasa and Wolff proved certain non-sharp $L^p$
    inequalities for a broader class of maximal functions arising from curves of
    the form $\{\Phi(x,\cdot)=r\}$, where $\Phi(x,y)$ satisfied Sogge's cinematic
    curvature condition. Under the additional hypothesis that $\Phi$ is algebraic,
    we obtain a sharp $L^3$ bound on the corresponding maximal function. Since the
    function $\Phi(x,y)=|x-y|$ is algebraic and satisfies the cinematic curvature
    condition, our result generalizes Wolff's $L^3$ bound.

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