  In this article we investigate a special class of non-doubling metric measure
spaces in order to describe the possible configurations of $P_{k,\rm s}^{\rm
c}$, $P_{k,\rm s}$, $P_{k,\rm w}^{\rm c}$ and $P_{k,\rm w}$, the sets of all $p
\in [1, \infty]$ for which the weak and strong type $(p,p)$ inequalities hold
for the centered and non-centered modified Hardy--Littlewood maximal operators,
$M^{\rm c}_k$ and $M_k$, $k \geq 1$. For any fixed $k$ we describe the
necessary conditions that $P_{k,\rm s}^{\rm c}$, $P_{k,\rm s}$, $P_{k,\rm
w}^{\rm c}$ and $P_{k,\rm w}$ must satisfy in general and illustrate each
admissible configuration with a properly chosen non-doubling metric measure
space. We also give some partial results related to an analogous problem stated
for varying $k$.
