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We consider the $L^p$ mapping properties of maximal averages associated tofamilies of curves, and thickened curves, in the plane. These include the(planar) Kakeya maximal function, the circular maximal functions of Wolff andBourgain, and their multi-parameter analogues. We propose a framework thatallows for a unified study of such maximal functions, and prove sharp $L^p\toL^p$ operator bounds in this setting. A key ingredient is an estimate fromdiscretized incidence geometry that controls the number of higher orderapproximate tangencies spanned by a collection of plane curves. We discussapplications to the F\"assler-Orponen restricted projection problem, and thedimension of Furstenberg-type sets associated to families of curves.

On Maximal Functions Associated to Families of Curves in the Plane