Maximal functions appear in many forms in harmonic analysis (an area of mathematics).
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.
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).
The following statements are central to the utility of the Hardy–Littlewood maximal operator.
[1] 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.
This can be easily proved by calculating Mχ, where χ 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 F defined on the upper-half plane and produces a function F* defined on Rn via the expression Observe that for a fixed x, the set
with vertex at (x,0) and axis perpendicular to the boundary of Rn.
Thus, the non-tangential maximal operator simply takes the supremum of the function F over a cone with vertex at the boundary of Rn.
One particularly important form of functions F 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 Φ on Rn such that and set for t > 0.
Such a result can be used to show that the harmonic extension of an Lp(Rn) 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.
is defined as for each x in Rn, where the supremum is taken over all balls B and
[2] The sharp function can be used to obtain a point-wise inequality regarding singular integrals.
Suppose we have an operator T which is bounded on L2(Rn), so we have for all smooth and compactly supported f. Suppose also that we can realize T as convolution against a kernel K in the sense that, whenever f and g are smooth and have disjoint support Finally we assume a size and smoothness condition on the kernel K: when
exists, many results that hold in the classical case (e.g. boundedness in