Maximal functions appear in many forms in harmonic analysis. 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.
The Hardy–Littlewood maximal function
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. 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.
If f ∈ L1, then there exists a csuch that, for all α > 0,
If f ∈ Lp, then Mf ∈ Lp and
Properties 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 for Mf requires a less elementary argument from geometric measure theory, such as the Vitali covering lemma. Property says the operator M is bounded on Lp; 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 for all other values of p can then be deduced from these two facts by an interpolation argument. It is worth noting does not hold for p = 1. This can be easily proved by calculating Mχ, where χ is the characteristic function of the unit ball centred at the origin.
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 is a cone in with vertex at 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.
Approximations of the identity
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. Then define One can show that and consequently obtain that converges to f in Lp for all 1 ≤ p < ∞. Such a result can be used to show that the harmonic extension of an Lp 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. Moreover, with some appropriate conditions on, one can get that
The sharp maximal function
For a locally integrable functionf on Rn, the sharp maximal function is defined as for each x in Rn, where the supremum is taken over all balls B and is the integral average of over the ball. 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, so we have for all smooth and compactly supportedf. 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. Then for a fixed r > 1, we have for all x in Rn.
Let be a probability space, and T : X → X a measure-preserving endomorphism of X. The maximal function of f ∈ L1 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.
Martingale Maximal Function
If is a martingale, we can define the martingale maximal function by. If exists, many results that hold in the classical case hold with respect to and.
