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.
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In their original paper, G.H. Hardy and J.E. Littlewood explained their maximal inequality in the language of cricket averages. Given a function defined on the uncentred Hardy–Littlewood maximal function of is defined as
at each . Here, the supremum is taken over balls in which contain the point and denotes the measure of (in this case a multiple of the radius of the ball raised to the power ). One can also study the centred maximal function, where the supremum is taken just over balls which have centre . In practice there is little difference between the two.
The following statements[1] are central to the utility of the Hardy–Littlewood maximal operator.
(a) For (), is finite almost everywhere.
(b) If , then there exists a such that, for all ,
(c) If (), then and
where depends only on and .
Properties (b) is called a weak-type bound of M(f). For an integrable function, it corresponds to the elementary Markov inequality; however, M(f) is never integrable, unless f is zero a.e., so that the proof of the weak bound (b) for M(f) requires a less elementary argument from geometric measure theory, such as the Vitali covering lemma. Property (c) says the operator is bounded on ; it is clearly true when , 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 can then be deduced from these two facts by an interpolation argument.
It is worth noting (c) does not hold for . This can be easily proved by calculating , where is the characteristic function of the unit ball centred at the origin.
The Hardy–Littlewood maximal operator appears in many places but some of its most notable uses are in the proofs of the Lebesgue differentiation theorem and Fatou's theorem and in the theory of singular integral operators.
The non-tangential maximal function takes a function defined on the upper-half plane and produces a function defined on via the expression
Obverse that for a fixed , the set is a cone in with vertex at and axis perpendicular to the boundary of . Thus, the non-tangential maximal operator simply takes the supremum of the function over a cone with vertex at the boundary of .
One particularly important form of functions in which study of the non-tangential maximal function is important is formed from an approximation to the identity. That is, we fix an integrable smooth function on such that and set
for . Then define
One can show[1] that
and consequently obtain that converges to in for all . Such a result can be used to show that the harmonic extension of an function to the upper-half plane converges non-tangentially to that function. More general results can be obtained where the Laplacian is replaced by an elliptic operator via similar techniques.
For a locally integrable function on , the sharp maximal function is defined as
for each , where the supremum is taken over all balls .[2]
The sharp function can be used to obtain a point-wise inequality regarding singular integrals. Suppose we have an operator which is bounded on , so we have
for all smooth and compactly supported . Suppose also that we can realise as convolution against a kernel in the sense that, whenever and are smooth and have disjoint support
Finally we assume a size and smoothness condition on the kernel :
when . Then for a fixed , we have
for all .[1]
Let be a probability space, and a measure-preserving endomorphism of . The maximal function of a function is
The maximal function of f verifies a weak bound analogous to the Hardy–Littlewood maximal inequality:
that is a restatement of the maximal ergodic theorem.