Maximal function
Maximal functions appear in many forms in harmonic analysis (an area of mathematics). One of the most important of these is the Hardy–Littlewood maximal function. They play an important role in understanding, for example, the differentiability properties of functions, singular integrals and partial differential equations. They often provide a deeper and more simplified approach to understanding problems in these areas than other methods.
In their original paper, G.H. Hardy and J.E. Littlewood explained their maximal inequality in the language of cricket averages. Given a function f defined on Rn, the uncentred Hardy–Littlewood maximal function Mf of f is defined as
at each x in Rn. Here, the supremum is taken over balls B in Rn which contain the point x and |B| denotes the measure of B (in this case a multiple of the radius of the ball raised to the power n). One can also study the centred maximal function, where the supremum is taken just over balls B which have centre x. In practice there is little difference between the two.
The following statements are central to the utility of the Hardy–Littlewood maximal operator.
Properties (b) is called a weak-type bound of Mf. For an integrable function, it corresponds to the elementary Markov inequality; however, Mf is never integrable, unless f = 0 almost everywhere, so that the proof of the weak bound (b) for Mf requires a less elementary argument from geometric measure theory, such as the Vitali covering lemma. Property (c) says the operator M is bounded on Lp(Rn); it is clearly true when p = ∞, since we cannot take an average of a bounded function and obtain a value larger than the largest value of the function. Property (c) for all other values of p can then be deduced from these two facts by an interpolation argument.
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