Spherical maximal operator on symmetric spaces, an end point estimate (Q1433183)

Spherical maximal operators \({\mathcal M}f\) were first studied on Euclidean spaces \({\mathbb R}^n\) by E. Stein, \(n\geq 3\), and J. Bourgain, \(n=2\). On symmetric spaces of rank one A. Nevo and E. Stein proved the \(L^p\)-boundedness for \(p>\frac{n}{n-1}\), \(n>3\). The result was extended by A. Ionescu for \(n=2\). It is known that the result does not hold for \(1\leq p\leq\frac{n}{n-1}\). The author regards the borderline case \(p=n'=\frac{n}{n-1}\) for radial functions. He proves that boundedness holds if we take a smaller source, namely a Lorentz space \(L^{n',1}\) and bigger target spaces, namely the Lorentz spaces \(L^{n',\infty}\). The theorem holds for \(n\geq 2\) and the proof uses geometry arguments and volume estimates.