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

TL;DR

This paper proves a sharp L^3 bound for a class of algebraic variable coefficient Wolff circular maximal functions, extending Wolff's classical result to a broader algebraic setting using computational geometry techniques.

Contribution

It introduces a new approach to maximal function bounds by leveraging algebraic properties and computational geometry methods, generalizing Wolff's L^3 estimate.

Findings

Established sharp L^3 bounds for algebraic variable coefficient Wolff maximal functions.

Extended Wolff's classical results to a broader algebraic class of functions.

Applied computational geometry techniques like vertical cell decompositions and random sampling.

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.