A simple proof of the boundedness of Bourgain’s circular maximal operator

Given a set $E=(0, \infty)$, the circular maximal operator $\mathcal{M}$ associated with the parameter set $E$ is defined as the supremum of the circular means of a function when the radii of the circles are in $E$. Using stationary phase method, we give a simple proof of the $L^p$, $p>2$ boundedness of Bourgain's circular maximal operator.