Convergence and divergence almost everywhere of spherical means for radial functions

Abstract:

Let $d > 1$. It will be shown that the maximal operator ${S^*}$ of spherical means ${S_R},R > 0$, is bounded on ${L^p}({{\mathbf {R}}^d})$ radial functions when $2d/(d + 1) < p < 2d/(d - 1)$, and it implies that, for every ${L^p}({{\mathbf {R}}^d})$ radial function $f(t),{S_R}f(t)$ converges to $f(t)$ for a.e. $t \in {{\mathbf {R}}^d}$ when $2d/(d + 1) < p \leq 2$. Also, it will be proved that there is an ${L^{2d/(d + 1)}}({R^d})$ radial function $f(t)$ with compact support such that ${S_R}f(t)$ diverges for a.e. $t \in {R^d}$.