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

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 $\{\mathrm{\Phi }(x,\cdot )=r\}$, where $\mathrm{\Phi }(x,y)$ satisfied Sogge’s cinematic curvature condition. Under the additional hypothesis that $\mathrm{\Phi }$ is algebraic, we obtain a sharp $L^3$ bound on the corresponding maximal function. Since the function $\mathrm{\Phi }(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ 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.