Hermite expansions on R(N) for radial functions

It is proved that the Riesz means S(R)(delta)f, delta > 0, for the Hermite expansions on R(n), n greater-than-or-equal-to 2, satisfy the uniform estimates \\S(R)(delta)f\\p less-than-or-equal-to C \\f\\p for all radial functions if and only if p lies in the interval 2n/(n + 1 + 2delta) < p < 2n/(n - 1 - 2delta).