We extend the estimates for maximal Fourier restriction operators proved by M\"{ u}ller, Ricci, and Wright in \cite{ MR3960255} and Ramos in \cite{ MR4055940} to the case of arbitrary convex curves in the plane, with constants uniform in the curve. The improvement over M\"{ u}ller, 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 in \cite{ MR1960918}. 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.
Categories: math.CA