| dbp:proof
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- For p = ∞, the inequality is trivial . For 1≤ p < ∞, we use the lemma.
For every x such that Mf > t, by definition, we can find a ball Bx centered at x such that
:
Thus {Mf > t} is a subset of the union of such balls, as x varies in {Mf > t}. This is trivial since x is contained in Bx. By the lemma, we can find, among such balls, a sequence of disjoint balls Bj such that the union of 5Bj covers {Mf > t}.
It follows:
:
This completes the proof of the weak-type estimate.
The Lp bounds for p > 1 can be deduced from the weak bound by the Marcinkiewicz interpolation theorem.
Here is how the argument goes in this particular case.
Define the function by if and 0 otherwise.
We have then
:
and, by the definition of maximal function
:
By the weak-type estimate applied to , we have:
:
Then
:
By the estimate above we have:
: (en)
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