A Simple Proof of the Boundedness of Bourgain’s Circular Maximal Operator

Abstract

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.