In mathematics, the Hardy–Littlewood maximal operator M is a significant non-linear operator used in real analysis and harmonic analysis. It takes a locally integrable function f : Rd → C and returns another function Mf that, at each point x ∈ Rd, gives the maximum average value that f can have on balls centered at that point. More precisely,
Contents
where B(x, r) is the ball of radius r centred at x, and |E| denotes the d-dimensional Lebesgue measure of E ⊂ Rd.
The averages are jointly continuous in x and r, therefore the maximal function Mf, being the supremum over r > 0, is measurable. It is not obvious that Mf is finite almost everywhere. This is a corollary of the Hardy–Littlewood maximal inequality
Hardy–Littlewood maximal inequality
This theorem of G. H. Hardy and J. E. Littlewood states that M is bounded as a sublinear operator from the Lp(Rd) to itself for p > 1. That is, if f ∈ Lp(Rd) then the maximal function Mf is weak L1-bounded and Mf ∈ Lp(Rd). Before stating the theorem more precisely, for simplicity, let {f > t} denote the set {x | f(x) > t}. Now we have:
Theorem (Weak Type Estimate). For d ≥ 1 and f ∈ L1(Rd), there is a constant Cd > 0 such that for all λ > 0, 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 (Strong Type Estimate). For d ≥ 1, 1 < p ≤ ∞, and f ∈ Lp(Rd),
there is a constant Cp,d > 0 such that
In the strong type estimate the best bounds for Cp,d are unknown. However subsequently Elias M. Stein used the Calderón-Zygmund method of rotations to prove the following:
Theorem (Dimension Independence). For 1 < p ≤ ∞ one can pick Cp,d = Cp independent of d.
Proof
While there are several proofs of this theorem, a common one is given below: For p = ∞, the inequality is trivial (since the average of a function is no larger than its essential supremum). For 1 < p < ∞, first we shall use the following version of the Vitali covering lemma to prove the weak-type estimate. (See the article for the proof of the lemma.)
Lemma. Let X be a separable metric space and
If Mf(x) > t, then, by definition, we can find a ball Bx centered at x such that
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. We next deduce from this the Lp bounds. Define b by b(x) = f(x) if |f(x)| > t/2 and 0 otherwise. By the weak-type estimate applied to b, we have:
with C = 5d. Then
By the estimate above we have:
where the constant Cp depends only on p and d. This completes the proof of the theorem.
Note that the constant
Applications
Some applications of the Hardy–Littlewood 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 ∈ L1(Rn) and
where
We write f = h + g where h is continuous and has compact support and g ∈ L1(Rn) 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
Discussion
It is still unknown what the smallest constants Cp,d and Cd 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 Cp,d on the dimension, that is, Cp,d = Cp for some constant Cp > 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 Bx are required to merely contain x, rather than be centered at x. There is also the dyadic HL maximal operator
where Qx ranges over all dyadic cubes containing the point x. Both of these operators satisfy the HL maximal inequality.