Stein's generalized spherical maximal operator

Abstract

This survey concerns Stein's analytic family of generalized spherical means and the associated maximal operator. We review Stein's original theorem, the sharp necessary conditions for boundedness, and the complete theory for radial functions. We then explain the connection between positive results and local smoothing estimates for the half-wave propagator and, more generally, for Fourier integral operators. This point of view yields the sharp two-dimensional result through the work of Guth, H. Wang, and R. Zhang and, when combined with the recent local smoothing theorems, an improved higher-dimensional range. As a byproduct, the maximal operator associated with the first radial derivative of spherical means is bounded on $L^p(\R^4)$ for $2<p<3$, a case left open in recent work of Kwaśnicki. We keep track of the relation between the local smoothing exponents and the resulting maximal estimates, and identify the region that remains open.