Maximal inequalities for derivatives of spherical means

Abstract

We give an alternative formulation of Stein's maximal inequality for generalised spherical averages in terms of derivatives of standard spherical means: if \[ k \ge 0, \qquad d \ge 2 k + 3 , \qquad \frac{d}{d - k - 1} < p < \frac{d - 1}{k} , \] and $\sigma$ is the normalised surface measure on the unit sphere $\mathbb S$, then the maximal operator \[f \mapsto \sup_{r > 0} \, \biggl\lvert r^k (\tfrac{d}{dr})^k \int_{\mathbb S} f(\cdot + r y) \sigma(dy) \biggr\rvert\] is bounded on $L^p$, with a constant that is independent of the dimension $d$.