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. It takes a locally integrable function f : R^d → C and returns another function Mf that, at each point x ∈ R^d, gives the maximum average value that f can have on balls centered at that point. More precisely,

where B(x, r) is the ball of radius r centred at x, and |E| denotes the d-dimensional Lebesgue measure of E ⊂ R^d.

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

This theorem of G. H. Hardy and J. E. Littlewood states that M is bounded as a sublinear operator from the L^p(R^d) to itself for p > 1. That is, if f ∈ L^p(R^d) then the maximal function Mf is weak L¹-bounded and Mf ∈ L^p(R^d). 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(R^d), there is a constant C_d > 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:

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