Given a set $E\subset (0,\infty)$, the spherical maximal
operator associated to the parameter set $E$ is defined as the
supremum of the spherical means of a function when the radii
of the spheres are in $E$. The aim of the paper is to study
boundedness properties of these operators on the spaces
$L^p(|x|^{\alpha})$. It is shown that the range of values of
$\alpha$ for which boundedness holds behaves essentially as
follows: (i) for $p > n/(n-1)$ and negative $\alpha$ the range
does not depend on $E$; (ii) when $\alpha$ is positive it
depends only on the Minkowski dimension of $E$; (iii) if $p
< n/(n-1)$ and $\alpha$ is negative, sets with the same
Minkowski dimension can give different ranges of
boundedness.