In mathematics, the HardyâÂÂLittlewood maximal operator M is a significant non-linear operator used in real analysis and harmonic analysis.
The operator takes a locally integrable function and returns another function , where is the supremum of the average of among all possible balls centered on . Formally,
where |E| denotes the d-dimensional Lebesgue measure of a subset E â R<sup>d</sup>, and is the ball of radius, , centered at the point .
Since is locally integrable, the averages are jointly continuous in x and r, so the maximal function Mf, being the supremum over r > 0, is measurable.
A nontrivial corollary of the HardyâÂÂLittlewood maximal inequality states that is finite almost everywhere for functions in .
This theorem of G. H. Hardy and J. E. Littlewood states that M is bounded as a sublinear operator from L<sup>p</sup>(R<sup>d</sup>) to itself for p > 1. That is, if f â L<sup>p</sup>(R<sup>d</sup>) then the maximal function Mf is weak L<sup>1</sup>-bounded and Mf â L<sup>p</sup>(R<sup>d</sup>). Before stating the theorem more precisely, for simplicity, let {f > t} denote the set {x | f(x) > t}. Now we have:
<blockquote>Theorem (Weak Type Estimate). For d âÂÂ¥ 1, there is a constant C<sub>d</sub> > 0 such that for all û > 0 and f â L<sup>1</sup>(R<sup>d</sup>), we have:
</blockquote>
With the HardyâÂÂLittlewood maximal inequality in hand, the following strong-type estimate is an immediate consequence of the Marcinkiewicz interpolation theorem:
<blockquote>Theorem (Strong Type Estimate). For d âÂÂ¥ 1, 1 < p ⤠âÂÂ, and f â L<sup>p</sup>(R<sup>d</sup>), there is a constant C<sub>p,d</sub> > 0 such that
</blockquote>
In the strong type estimate the best bounds for C<sub>p,d</sub> are unknown. However subsequently Elias M. Stein used the Calderón-Zygmund method of rotations to prove the following:
<blockquote>Theorem (Dimension Independence). For 1 < p ⤠â one can pick C<sub>p,d</sub> = C<sub>p</sub> independent of d.</blockquote>
While there are several proofs of this theorem, a common one is given below, that uses the following version of the Vitali covering lemma to prove the weak-type estimate. (See the article for the proof of the lemma.)
Note that the constant in the proof can be improved to by using the inner regularity of the Lebesgue measure, and the finite version of the Vitali covering lemma. See the Discussion section below for more about optimizing the constant.
Some applications of the HardyâÂÂLittlewood Maximal Inequality include proving the following results:
Here we use a standard trick involving the maximal function to give a quick proof of Lebesgue differentiation theorem. (But remember that in the proof of the maximal theorem, we used the Vitali covering lemma.) Let f â L<sup>1</sup>(R<sup>n</sup>) and
where
We write f = h + g where h is continuous and has compact support and g â L<sup>1</sup>(R<sup>n</sup>) with norm that can be made arbitrary small. Then
by continuity. Now, ég ⤠2Mg and so, by the theorem, we have:
Now, we can let and conclude éf = 0 almost everywhere; that is, exists for almost all x. It remains to show the limit actually equals f(x). But this is easy: it is known that (approximation of the identity) and thus there is a subsequence almost everywhere. By the uniqueness of limit, f<sub>r</sub> â f almost everywhere then.
It is still unknown what the smallest constants C<sub>p,d</sub> and C<sub>d</sub> are in the above inequalities. However, a result of Elias Stein about spherical maximal functions can be used to show that, for 1 < p < âÂÂ, we can remove the dependence of C<sub>p,d</sub> on the dimension, that is, C<sub>p,d</sub> = C<sub>p</sub> for some constant C<sub>p</sub> > 0 only depending on p. It is unknown whether there is a weak bound that is independent of dimension.
There are several common variants of the Hardy-Littlewood maximal operator which replace the averages over centered balls with averages over different families of sets. For instance, one can define the uncentered HL maximal operator (using the notation of Stein-Shakarchi)
where the balls B<sub>x</sub> are required to merely contain x, rather than be centered at x. There is also the dyadic HL maximal operator
where Q<sub>x</sub> ranges over all dyadic cubes containing the point x. Both of these operators satisfy the HL maximal inequality.