In 1997, Thomas Wolff proved sharp bounds for his circular maximal function, and in 1999, Kolasa and Wolff proved certain non-sharp inequalities for a broader class of maximal functions arising from curves of the form , where satisfied Sogge’s cinematic curvature condition. Under the additional hypothesis that is algebraic, we obtain a sharp bound on the corresponding maximal function. Since the function is algebraic and satisfies the cinematic curvature condition, our result generalizes Wolff’s 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.
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Joshua Zahl, estimates for an algebraic variable coefficient Wolff circular maximal function. Rev. Mat. Iberoam. 28 (2012), no. 4, pp. 1061–1090DOI 10.4171/RMI/703