In mathematics, the Hardy–Littlewood maximal operator M is a significant non-linear operator used in real analysis and harmonic analysis.
For any point x ∈ Rd, the function Mf returns the maximum of a set of reals, namely the set of average values of f for all the balls B(x, r) of any radius r at x.
Formally, where |E| denotes the d-dimensional Lebesgue measure of a subset E ⊂ Rd.
The averages are jointly continuous in x and r, so the maximal function Mf, being the supremum over r > 0, is measurable.
This theorem of G. H. Hardy and J. E. Littlewood states that M is bounded as a sublinear operator from Lp(Rd) to itself for p > 1.
Before stating the theorem more precisely, for simplicity, let {f > t} denote the set {x | f(x) > t}.
For d ≥ 1, there is a constant Cd > 0 such that for all λ > 0 and f ∈ L1(Rd), 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).
[1] 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.[1][2]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.
a family of open balls with bounded diameter.
By the lemma, we can find, among such balls, a sequence of disjoint balls Bj such that the union of 5Bj covers {Mf > t}.
The Lp bounds for p > 1 can be deduced from the weak
by using the inner regularity of the Lebesgue measure, and the finite version of the Vitali covering lemma.
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.
It remains to show the limit actually equals f(x).
It is still unknown what the smallest constants Cp,d and Cd are in the above inequalities.
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.