Maximal Operator
   HOME

TheInfoList



OR:

Maximal functions appear in many forms in
harmonic analysis Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...
(an area of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
). One of the most important of these is the
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. Definition The operator takes a locally integrable function f: \R^d \to \mathbb C and returns another ...
. 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 John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanu ...
explained their maximal inequality in the language of
cricket Cricket is a Bat-and-ball games, bat-and-ball game played between two Sports team, teams of eleven players on a cricket field, field, at the centre of which is a cricket pitch, pitch with a wicket at each end, each comprising two Bail (cr ...
averages. Given a function ''f'' defined on R''n'', the uncentred Hardy–Littlewood maximal function ''Mf'' of ''f'' is defined as :(Mf)(x) = \sup_ \frac \int_B , f, at each ''x'' in R''n''. Here, the supremum is taken over balls ''B'' in R''n'' 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.


Basic properties

The following statements are central to the utility of the Hardy–Littlewood maximal operator. * (a) For ''f'' ∈ ''Lp''(R''n'') (1 ≤ ''p'' ≤ ∞), ''Mf'' is finite almost everywhere. * (b) If ''f'' ∈ ''L1''(R''n''), then there exists a ''c'' such that, for all α > 0, ::, \, \leq \frac\int_ , f, . *(c) If ''f'' ∈ ''Lp''(R''n'') (1 < ''p'' ≤ ∞), then ''Mf'' ∈ ''Lp''(R''n'') and ::\, Mf\, _ \leq A \, f\, _, :where ''A'' depends only on ''p'' and ''c''. Properties (b) is called a weak-type bound of ''Mf''. For an integrable function, it corresponds to the elementary
Markov inequality In probability theory, Markov's inequality gives an upper bound on the probability that a non-negative random variable is greater than or equal to some positive constant. Markov's inequality is tight in the sense that for each chosen positive con ...
; 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 In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The co ...
. Property (c) says the operator ''M'' is bounded on ''Lp''(R''n''); 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. It is worth noting (c) does not hold for ''p'' = 1. This can be easily proved by calculating ''M''χ, where χ is the characteristic function of the unit ball centred at the origin.


Applications

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 In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limiting average taken around the point. The theorem is named for Henri Lebesgu ...
and
Fatou's theorem In mathematics, specifically in complex analysis, Fatou's theorem, named after Pierre Fatou, is a statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk. Motivation and statement of t ...
and in the theory of
singular integral operators In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator : T(f)(x) = \int K(x,y)f(y) \, dy, wh ...
.


Non-tangential maximal functions

The non-tangential maximal function takes a function ''F'' defined on the upper-half plane :\mathbf^_+ := \left \ and produces a function ''F*'' defined on R''n'' via the expression :F^*(x) = \sup_ , F(y,t), . Observe that for a fixed ''x'', the set \ is a cone in \mathbf^_+ with vertex at (''x'',0) and axis perpendicular to the boundary of R''n''. Thus, the non-tangential maximal operator simply takes the supremum of the function ''F'' over a cone with vertex at the boundary of R''n''.


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 R''n'' such that :\int_ \Phi = 1 and set :\Phi_t(x) = t^ \Phi(\tfrac) for ''t'' > 0. Then define :F(x,t) = f \ast \Phi_t(x) := \int_ f(x-y)\Phi_t(y) \, dy. One can show that :\sup_, f \ast \Phi_t(x), \leq (Mf)(x) \int_ \Phi and consequently obtain that f \ast \Phi_t(x) converges to ''f'' in ''Lp''(R''n'') for all 1 ≤ ''p'' < ∞. Such a result can be used to show that the harmonic extension of an ''Lp''(R''n'') 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 \Phi, one can get that :F^*(x) \leq C(Mf)(x).


The sharp maximal function

For a locally integrable function ''f'' on R''n'', the sharp maximal function f^\sharp is defined as : f^\sharp(x) = \sup_ \frac \int_B , f(y) - f_B, \, dy for each ''x'' in R''n'', where the supremum is taken over all balls ''B'' and f_B is the integral average of f over the ball B. The sharp function can be used to obtain a point-wise inequality regarding
singular integrals In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator : T(f)(x) = \int K(x,y)f(y) \, dy, wh ...
. Suppose we have an operator ''T'' which is bounded on ''L2''(R''n''), so we have : \, T(f)\, _ \leq C\, f\, _, 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 : \int g(x) T(f)(x) \, dx = \iint g(x) K(x-y) f(y) \, dy\,dx. Finally we assume a size and smoothness condition on the kernel ''K'': : , K(x-y)-K(x), \leq C \frac, when , x, \geq 2, y, . Then for a fixed ''r'' > 1, we have : (T(f))^\sharp(x) \leq C(M(, f, ^r))^\frac(x) for all ''x'' in R''n''.


Maximal functions in ergodic theory

Let (X,\mathcal,m) be a probability space, and ''T'' : ''X'' → ''X'' a measure-preserving endomorphism of ''X''. The maximal function of ''f'' ∈ ''L1''(''X'',''m'') is :f^*(x):=\sup_\frac\sum_i^, f(T^i(x)), . The maximal function of ''f'' verifies a weak bound analogous to the Hardy–Littlewood maximal inequality: : m\left(\\right)\leq\frac, that is a restatement of the maximal ergodic theorem.


Martingale Maximal Function

If \ is a martingale, we can define the martingale maximal function by f^*(x) = \sup_n, f_n(x), . If f(x) = \lim_f_n(x) exists, many results that hold in the classical case (e.g. boundedness in L^p, 1 and the weak L^1 inequality) hold with respect to f and f^*.{{cite book , last = Stein , first = Elias M. , title = Topics in Harmonic Analysis Related to the Littlewood-Paley Theory , chapter = Chapter IV: The General Littlewood-Paley Theory, publisher = Princeton University Press , place = Princeton, New Jersey, date = 2004


References

*L. Grafakos, ''Classical and Modern Fourier Analysis'', Pearson Education, Inc., New Jersey, 2004 *E.M. Stein, ''Harmonic Analysis'', Princeton University Press, 1993 *E.M. Stein, ''Singular Integrals and Differentiability Properties of Functions'', Princeton University Press, 1971 *E.M. Stein, ''Topics in Harmonic Analysis Related to the Littlewood-Paley Theory'', Princeton University Press, 1970


Notes

Real analysis