JournalsrmiVol. 28, No. 4pp. 1061–1090

L3L^3 estimates for an algebraic variable coefficient Wolff circular maximal function

  • Joshua Zahl

    The University of British Columbia, Vancouver, USA
$L^3$ estimates for an algebraic variable coefficient Wolff circular maximal function cover
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Abstract

In 1997, Thomas Wolff proved sharp L3L^3 bounds for his circular maximal function, and in 1999, Kolasa and Wolff proved certain non-sharp LpL^p inequalities for a broader class of maximal functions arising from curves of the form {Φ(x,)=r}\{\Phi(x,\cdot)=r\}, where Φ(x,y)\Phi(x,y) satisfied Sogge’s cinematic curvature condition. Under the additional hypothesis that Φ\Phi is algebraic, we obtain a sharp L3L^3 bound on the corresponding maximal function. Since the function Φ(x,y)=xy\Phi(x,y)=|x-y| is algebraic and satisfies the cinematic curvature condition, our result generalizes Wolff’s L3L^3 bound. The algebraicity condition allows us to employ the techniques of vertical cell decompositions and random sampling, which have been extensively developed in the computational geometry literature.

Cite this article

Joshua Zahl, L3L^3 estimates for an algebraic variable coefficient Wolff circular maximal function. Rev. Mat. Iberoam. 28 (2012), no. 4, pp. 1061–1090

DOI 10.4171/RMI/703