  We study a generalized spherical means operator, viz. generalized spherical
mean Radon transform, acting on radial functions. As the main results, we find
conditions for the associated maximal operator and its local variant to be
bounded on power weighted Lebesgue spaces. This translates, in particular, into
almost everywhere convergence to radial initial data results for solutions to
certain Cauchy problems for classical Euler-Poisson-Darboux and wave equations.
Moreover, our results shed some new light to the interesting and important
question of optimality of the yet known $L^p$ boundedness results for the
maximal operator in the general non-radial case. It appears that these could
still be notably improved, as indicated by our conjecture of the ultimate sharp
result.
