User:Potahto/Maximal functions

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.

The Hardy-Littlewood maximal function

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G.H. Hardy was the first to consider maximal functions in the hope of better understanding cricket scores. 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 are have centre  . In practice there is little difference between the two.

Basic properties

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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 and (c) says the operator   is bounded on  . Property (b) can be proved using the Vitali covering lemma. Property (c) 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.

Applications

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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.

Non-tangential maximal functions

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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  .

Approximations of the identity

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One particularly important form of functions   in which study of the non-tangential maximal function is important is 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.

The sharp maximal function

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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]

References

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  • 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 & G. Weiss, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, 1971

Notes

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  1. ^ a b c Stein, Elias (1993). "Harmonic Analysis". Princeton University Press. {{cite news}}: Check date values in: |date= (help); Cite has empty unknown parameter: |coauthors= (help)
  2. ^ Grakakos, Loukas (2004). "7". Classical and Modern Fourier Analysis. New Jersey: Pearson Education, Inc. {{cite book}}: Check date values in: |date= (help); Cite has empty unknown parameter: |coauthors= (help)