  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.
