Maximal inequalities for derivatives of spherical means

We give an alternative formulation of Stein’s maximal inequality for generalised spherical averages in terms of derivatives of standard spherical means: if $$k\ge 0,d\ge 2k+3,\frac{d}{d-k-1}<p<\frac{d-1}{k},$$ and $\sigma$ is the normalised surface measure on the unit sphere $\mathbb{S}$, then the maximal operator $$f\mapsto \underset{r>0}{\sup }\mid r^k(\frac{d}{dr}{)}^k{\int }_{\mathbb{S}}f(\cdot +ry)\sigma (dy)\mid$$ is bounded on $L^p$, with a constant that is independent of the dimension $d$.