$L^3$ estimates for an algebraic variable coefficient Wolff circular maximal function
Joshua Zahl · Revista Matemática Iberoamericana · 2012
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.