Uniform maximal Fourier restriction for convex curves

Abstract

We extend the estimates for maximal Fourier restriction operators proved by Müller et al. (Rev Mat Iberoam 35:693–702, 2019) and Ramos (Proc Am Math Soc 148:1131–1138, 2020) to the case of arbitrary convex curves in the plane, with constants uniform in the curve. The improvement over Müller, Ricci, and Wright and Ramos is given by the removal of the \({\mathcal {C}}^2\) regularity condition on the curve. This requires the choice of an appropriate measure for each curve, that is suggested by an affine invariant construction of Oberlin (Michigan Math J 51:13–26, 2003). As corollaries, we obtain a uniform Fourier restriction theorem for arbitrary convex curves and a result on the Lebesgue points of the Fourier transform on the curve.