# Revista Matemática Iberoamericana

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**Volume 28, Issue 4, 2012, pp. 1061–1090**

**DOI: 10.4171/RMI/703**

Published online: 2012-10-14

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

Joshua Zahl^{[1]}(1) The University of British Columbia, Vancouver, USA

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. 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.

*Keywords: *Wolff circular maximal function, Besicovitch–Radó–Kinney set, vertical algebraic decomposition

Zahl Joshua: $L^3$ estimates for an algebraic variable coefficient Wolff circular maximal function. *Rev. Mat. Iberoamericana* 28 (2012), 1061-1090. doi: 10.4171/RMI/703