The spherical maximal operator on radial functions (Q655469)

The spherical maximal function in \(\mathbb{R}^n\) is bounded in \(L^p\) if and only if \(p> n/(n-1)\). The case \(n=2\) is more difficult then \(n\geq 3\). In the present work the action of the spherical maximal function is restricted to radial functions. Then it is itself radial. Even now it is unbounded in \(L^p\) is \(p\leq n/(n-1)\). The authors use one-dimensional auxiliary operators to obtain pointwise bounds in the radial case. Sharp inequalities with weights are given. Also an inequality of the Fefferman-Stein type is provided. The introduction of this well written paper contains a goodl overview.