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

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.