When I was reading Singular Integrals and Related Topics by ShanHen Lu, Yong Ding and Duncan yan(ISBN-13:978-981-270-623-2), I noticed that in page 2, after the author proved that several Hardy-Littlewood maximal functions are pointsise equivalent to each other and is labeled as formula (1.1.4):
$$ C_0Mf(x) \leq C_1 M’f(x) \leq C_2 M’’f(x) \leq C_3Mf(x) $$
and when it’s time to prove that Hardy-Littlewood maximal functions are lower semi-continuous, he mentioned:
By (1.1.4), we only need to show it for $M’f(x)$ where $M’f$ is just one type of the Hardy-Littlewood maximal functions.
My question is that: after I have proved that $M’f(x)$ is lower semi-continuous, how can I prove that the other types of Hardy-Littlewood maximal functions are also lower semi-continuous?
If $Mf(x) = \lambda M’f(x)$, then it’s easy to prove just by the definition of lower semi-continuous, since the set required of the two functions have some equivalent relation, but this is not the case when the fact is that as in (1.1.4) I couldn’t “convert” the set, but only to get a “set inclusion”.
I wonder the ideas behind the “only need to show it for $M’f(x)$”, thanks a lot for any help.