  In this paper, we investigate sharp damping estimates for a class of one
dimensional oscillatory integral operators with real-analytic phases. By
establishing endpoint estimates for suitably damped oscillatory integral
operators, we are able to give a new proof of the sharp $L^p$ estimates which
have been proved by Xiao in Endpoint estimates for one-dimensional oscillatory
integral operators, \emph{Advances in Mathematics}, \textbf{316}, 255-291
(2017). The damping estimates obtained in this paper are of independent
interest.

Classical Analysis and ODEs