The spherical ergodic theorem revisited (Q1028224)

The pointwise ergodic theorem for averages over spheres in a measure-preserving \(\mathbb R^d\)-action and functions in \(L^p\) was shown for \(d\geq 3\) and \(p\geq d/(d-1)\) by \textit{R. L. Jones} [J. Anal. Math. 61, 29--45 (1993; Zbl 0828.28007)] and for \(d\geq2\) and \(p\geq d/(d-1)\) by \textit{M. T. Lacey} [J. Anal. Math. 67, 199--206 (1995; Zbl 0874.28021)]. Here a different proof of the theorem for \(d\geq3\) is given, by modifying \textit{J. L. Rubio de Francia}'s proof [Duke Math. J. 53, 395--404 (1986; Zbl 0612.42008)] of the spherical maximal theorem of \textit{E. M. Stein} [Proc. Natl. Acad. Sci. USA 73, 2174--2175 (1976; Zbl 0332.42018)].