For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$? (This is essentially a continuation of my previous question, here.)
Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further assume (though you can remove this assumption if you like) that each ball has positive finite volume. Let $M$ be the uncentred Hardy-Littlewood maximal operator, given by
$$Mf(x) := \sup_{B \ni x} \frac{1}{\mu(B)} \int_B |f(y)| \; d\mu(y)$$
with the supremum taken over balls containing $x$.
In my previous question, I asked when $M$ is not of weak type $(1,1)$, and was presented with an example ($\mathbb{R}^2$ with Gaussian measure) which isn't of weak type $(1,1)$, but which is of strong type $(p,p)$ for all $p>1$. (See Sjögren ('83), and Forzani, Scotto, Sjögren, Urbina ('02).)
Now I'm interested in the following question: are there any metric measure spaces $(X,d,\mu)$ for which $M$ is of strong type $(p,p)$ if and only if $p > p_0$ (or $p \geq p_0$) for some $p_0 > 1$ (note the strict inequality here)? In other words, do we ever have some but not all midpoint strong type estimates?
I know that there are maximal operators out there for which this is true, but I'm specifically interested in the uncentred Hardy-Littlewood maximal operator (and to some extent the centred version, though we have to assume some regularity of the measure here to ensure measurability).
 A: For the uncentred maximal function, one can cook up examples by using the star graph with a suitable weight on the vertices.  In more detail, if one takes a star graph (with the graph metric) with $n$ spokes, with the root vertex having measure one and the leaves having measure $n^{-1/p_0}$, then the uncentred maximal function has bounded $L^p$ operator norm for $p \geq p_0$ but unbounded norm for $p > p_0$, by testing this function on the indicator function of the root vertex.  If one then takes a disjoint union of such star graphs with the $n$ parameter going to infinity, one gets an example of the form you wanted.
For the centred maximal function, the star graph construction is no longer useful, but there are other, more complicated, constructions (based on metrics in vector spaces over finite fields) in Section 6 of my paper with Assaf Naor which can probably be adapted to give a suitable counterexample, though this would require a bit of tinkering (the examples there were designed to have weak (1,1) fail but strong (p,p) for all $p>1$).
A: It was proved by Alex Ionescu that the uncentered Hardy Littlewood maximal function is restricted weak type $(2,2)$ and not strong type $(p,p)$ for $p<2$ for hyperbolic spaces in http://arxiv.org/pdf/math/0007200v1.pdf
