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

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

2010-12-03

arxiv.org/abs/1012.0649v2

Mathematics

Classical Analysis and ODEs

25 pages, 2 figures. Typos corrected and other minor changes in exposition. To appear in Revista Mat. Iber

Scientific paper

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.

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