  We investigate the magnitude relation of the non-centered Hardy-Littlewood
maximal operators and centered one. By using a discretization technique, we
prove two facts: the first one is that the space is ultrametric if and only if
the two maximal operators are identical for all discrete measure; the second
is, the uncentred maximal operator is strictly greater than the centered one if
$(M,d_g)$ is a Riemannian manifold and $\mu$ is the Riemannian volume measure.

Classical Analysis and ODEs;Metric Geometry