Hardy–Littlewood maximal function

In mathematics, the Hardy–Littlewood maximal operator M is a significant non-linear operator used in real analysis and harmonic analysis.

Definition

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 ⊂ R^d, 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 Hardy–Littlewood maximal inequality states that is finite almost everywhere for functions in.

Hardy–Littlewood maximal inequality

This theorem of G. H. Hardy and J. E. Littlewood states that M is bounded as a sublinear operator from L^p(R^d) to itself for p > 1. That is, if f ∈ L^p then the maximal function Mf is weak L¹-bounded and Mf ∈ L^p. Before stating the theorem more precisely, for simplicity, let denote the set. Now we have:

Theorem. For d ≥ 1, there is a constant C_d > 0 such that for all λ > 0 and f ∈ L¹, we have:

With the Hardy–Littlewood maximal inequality in hand, the following strong-type estimate is an immediate consequence of the Marcinkiewicz interpolation theorem:

Theorem. For d ≥ 1, 1 < p ≤ ∞, and f ∈ L^p,
there is a constant C_p,d > 0 such that

In the strong type estimate the best bounds for C_p,d are unknown. However subsequently Elias M. Stein used the Calderón-Zygmund method of rotations to prove the following:

Theorem. For 1 < p ≤ ∞ one can pick C_p,d = C_p independent of d.

Proof

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.
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.

Applications

Some applications of the Hardy–Littlewood Maximal Inequality include proving the following results:

Lebesgue differentiation theorem
Rademacher differentiation theorem
Fatou's theorem on nontangential convergence.
Fractional integration theorem

Here we use a standard trick involving the maximal function to give a quick proof of Lebesgue differentiation theorem. Let f ∈ L¹ and
where
We write f = h + g where h is continuous and has compact support and g ∈ L¹ with norm that can be made arbitrary small. Then
by continuity. Now, Ωg ≤ 2Mg and so, by the theorem, we have:
Now, we can let and conclude Ωf = 0 almost everywhere; that is, exists for almost all x. It remains to show the limit actually equals f. But this is easy: it is known that and thus there is a subsequence almost everywhere. By the uniqueness of limit, f_r → f almost everywhere then.

Discussion

It is still unknown what the smallest constants C_p,d and C_d are in the above inequalities. However, a result of Elias Stein about spherical maximal functions can be used to show that, for 1 < p < ∞, we can remove the dependence of C_p,d on the dimension, that is, C_p,d = C_p for some constant C_p > 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
where the balls B_x are required to merely contain x, rather than be centered at x. There is also the dyadic HL maximal operator
where Q_x ranges over all dyadic cubes containing the point x. Both of these operators satisfy the HL maximal inequality.