The Spherical Maximal Function: Classical and Geometric Approaches

This thesis explores the Spherical Maximal Function and its associated Maximal Theorem, contrasting classical analytical methods with a geometric perspective. It begins by establishing the necessary harmonic analysis framework, focusing particularly on the decay properties of oscillatory integrals. The first two chapters of the thesis detail Stein's classical proof for dimensions $d \ge 3$, emphasizing the use of the Littlewood-Paley decomposition, square functions for $L^2$ bounds, and weak (1,1) estimates. In Chapter 3, the focus shifts to a purely geometric perspective. A series of key reductions and ideas are used to prove the $L^2$ bound for the local version of the operator for $d \ge 3$, relying solely on intersecting annuli. Finally, in Chapter 4, geometric intuition and Fourier analytic techniques are combined to provide an alternative proof of Stein’s theorem for the local version of the operator. The last section explores the geometric obstacles that arise when analyzing the problem in the plane understanding the first steps of Bourgain's proof of the circular maximal theorem.