What is 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 → C and returns another function Mf that, at each point x ∈ R, gives the maximum average value that f can have on balls centered at that point. More precisely,

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

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.