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

### Joshua Zahl

The University of British Columbia, Vancouver, USA

## 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. 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, $L^3$ 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