en>Jaan Vajakas at 21:23, 22 May 2013

2013-05-22T21:23:10Z

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In [[mathematics]], the '''Hardy–Littlewood maximal operator''' ''M'' is a significant non-linear [[Operation (mathematics)|operator]] used in [[real analysis]] and [[harmonic analysis]]. It takes a [[locally integrable]] function ''f'' : '''R'''''d'' → '''C''' and returns another function ''Mf'' that, at each point ''x'' ∈ '''R'''''d'', gives the maximum [[average|average value]] that ''f'' can have on balls centered at that point. More precisely,

:<math> Mf(x)=\sup_{r>0} \frac{1}{|B(x, r)|}\int_{B(x, r)} |f(y)|\, dy </math>

where ''B''(''x'', ''r'') is the ball of radius ''r'' centred at ''x'', and |''E''| denotes the [[Lebesgue measure|''d''-dimensional Lebesgue measure]] of ''E'' ⊂ '''R'''''d''.

The averages are jointly [[Continuous function|continuous]] in ''x'' and ''r'', therefore the maximal function ''Mf'', being the supremum over ''r'' > 0, is [[Measurable function|measurable]]. It is not obvious that ''Mf'' is finite almost everywhere. This is a corollary of the '''Hardy–Littlewood maximal inequality'''

==Hardy–Littlewood maximal inequality==
This theorem of [[G. H. Hardy]] and [[J. E. Littlewood]] states that ''M'' is [[bounded operator|bounded]] as a [[sublinear operator]] from the [[Lp space|''Lp''('''R'''''d'')]] to itself for ''p'' > 1. That is, if ''f'' ∈ ''Lp''('''R'''''d'') then the maximal function ''Mf'' is weak ''L''1-bounded and ''Mf'' ∈ ''Lp''('''R'''''d''). Before stating the theorem more precisely, for simplicity, let {''f'' > ''t''} denote the set {''x'' | ''f''(''x'') > ''t''}. Now we have:

<blockquote>'''Theorem (Weak Type Estimate).''' For ''d'' ≥ 1 and ''f'' ∈ ''L''1('''R'''''d''), there is a constant ''Cd'' > 0 such that for all λ > 0, we have:

:<math>\left |\{Mf > \lambda\} \right |< \frac{C_d}{\lambda} \Vert f\Vert_{L^1 (\mathbf{R}^d)}.</math>
</blockquote>

With the Hardy–Littlewood maximal inequality in hand, the following ''strong-type'' estimate is an immediate consequence of the [[Marcinkiewicz theorem|Marcinkiewicz interpolation theorem]]:

<blockquote>'''Theorem (Strong Type Estimate).''' For ''d'' ≥ 1, 1 < ''p'' ≤ ∞, and ''f'' ∈ ''Lp''('''R'''''d''),
there is a constant ''Cp,d'' > 0 such that

:<math> \Vert Mf\Vert_{L^p (\mathbf{R}^d)}\leq C_{p,d}\Vert f\Vert_{L^p(\mathbf{R}^d)}.</math>
</blockquote>

In the strong type estimate the best bounds for ''Cp,d'' are unknown.<ref name=Tao/> However subsequently [[Elias M. Stein]] used the Calderón-Zygmund method of rotations to prove the following:

<blockquote>'''Theorem (Dimension Independence).''' For 1 < ''p'' ≤ ∞ one can pick ''Cp,d'' = ''Cp'' independent of ''d''.<ref name=Tao>{{cite web|last=Tao|first=Terence|title=Stein’s spherical maximal theorem|url=http://terrytao.wordpress.com/2011/05/21/steins-spherical-maximal-theorem/|work=What's New|accessdate=22 May 2011}}</ref><ref name=Stein82>{{cite journal|last=Stein|first=E. M.|title=The development of square functions in the work of A. Zygmund.|journal=Bulletin of the American Mathematical Society New Series|date=S 1982|volume=7|issue=2|pages=359–376}}</ref></blockquote>

==Proof==
While there are several proofs of this theorem, a common one is given below: For ''p'' = ∞, the inequality is trivial (since the average of a function is no larger than its [[essential supremum]]). For 1 < ''p'' < ∞, first we shall use the following version of the [[Vitali covering lemma]] to prove the weak-type estimate. (See the article for the proof of the lemma.)

<blockquote>'''Lemma.''' Let ''X'' be a separable metric space and <math>\mathcal{F}</math> family of open balls with bounded diameter. Then <math>\mathcal{F}</math> has a countable subfamily <math>\mathcal{F}'</math> consisting of disjoint balls such that
:<math>\bigcup_{B \in \mathcal{F}} B \subset \bigcup_{B \in \mathcal{F'}} 5B</math>
where 5''B'' is ''B'' with 5 times radius.</blockquote>

If ''Mf''(''x'') > ''t'', then, by definition, we can find a ball ''Bx'' centered at ''x'' such that
:<math>\int_{B_x} |f|dy > t|B_x|.</math>
By the lemma, we can find, among such balls, a sequence of disjoint balls ''Bj'' such that the union of 5''Bj'' covers {''Mf'' > ''t''}.
It follows:
:<math>|\{Mf > t\}| \le 5^d \sum_j |B_j| \le {5^d \over t} \int |f|dy.</math>
This completes the proof of the weak-type estimate. We next deduce from this the ''Lp'' bounds. Define ''b'' by ''b''(''x'') = ''f''(''x'') if |''f''(''x'')| > ''t''/2 and 0 otherwise. By the weak-type estimate applied to ''b'', we have:
:<math>|\{Mf > t\}| \le {2C \over t} \int_{|f| > \frac{t}{2}} |f|dx, </math>
with ''C'' = 5''d''. Then
:<math>\|Mf\|_p^p = \int \int_0^{Mf(x)} pt^{p-1} dt dx = p \int_0^\infty t^{p-1} |\{ Mf > t \}| dt</math>
By the estimate above we have:
:<math>\|Mf\|_p^p \leq p \int_0^\infty t^{p-1} \left ({2C \over t} \int_{|f| > \frac{t}{2}} |f|dx \right ) dt = 2C p \int_0^\infty \int_{|f| > \frac{t}{2}} t^{p-2} |f| dx dt = C_p \|f\|_p^p</math>
where the constant ''Cp'' depends only on ''p'' and ''d''. This completes the proof of the theorem.

==Applications==
Some applications of the Hardy–Littlewood Maximal Inequality include proving the following results:
* [[Lebesgue differentiation theorem]]
* [[Rademacher's theorem|Rademacher differentiation theorem]]
* [[Fatou's theorem]] on nontangential convergence.
* [[Fractional integration theorem]]

Here we use a standard trick involving the maximal function to give a quick proof of Lebesgue differentiation theorem. (But remember that in the proof of the maximal theorem, we used the Vitali covering lemma.) Let ''f'' ∈ ''L''1('''R'''''n'') and

:<math>\Omega f (x) = \limsup_{r \to 0} f_r(x) - \liminf_{r \to 0} f_r(x)</math>

where

:<math>f_r(x) = \frac{1}{|B(x, r)|} \int_{B(x, r)} f(y) dy.</math>

We write ''f'' = ''h'' + ''g'' where ''h'' is continuous and has compact support and ''g'' ∈ ''L''1('''R'''''n'') with norm that can be made arbitrary small. Then

:<math>\Omega f \le \Omega g + \Omega h = \Omega g</math>

by continuity. Now, Ω''g'' ≤ 2''Mg'' and so, by the theorem, we have:

:<math>\left | \{ \Omega g > \varepsilon \} \right | \le \frac{2A}{\varepsilon} \|g\|_1</math>

Now, we can let <math>\|g\|_1 \to 0</math> and conclude Ω''f'' = 0 almost everywhere; that is, <math>\lim_{r \to 0} f_r(x)</math> exists for almost all ''x''. It remains to show the limit actually equals ''f''(''x''). But this is easy: it is known that <math>\|f_r - f\|_1 \to 0</math> ([[approximation of the identity]]) and thus there is a subsequence <math>f_{r_k} \to f</math> almost everywhere. By the uniqueness of limit, ''fr'' → ''f'' almost everywhere then.

==Discussion==
It is still unknown what the smallest constants ''Cp,d'' and ''Cd'' are in the above inequalities. However, a result of [[Elias Stein]] about spherical maximal functions can be used to show that, for 1 < ''p'' < ∞, we can remove the dependence of ''Cp,d'' on the dimension, that is, ''Cp,d'' = ''Cp'' for some constant ''Cp'' > 0 only depending on ''p''. It is unknown whether there is a weak bound that is independent of dimension.

There are several common variants of the Hardy-Littlewood maximal operator which replace the averages over centered balls with averages over different families of sets. For instance, one can define the ''uncentered'' HL maximal operator (using the notation of Stein-Shakarchi)

:<math> f^*(x) = \sup_{x \in B_x} \frac{1}{|B_x|} \int_{B_x} |f(y)| dy</math>

where the balls ''Bx'' are required to merely contain x, rather than be centered at x. There is also the ''dyadic'' HL maximal operator

:<math>M_\Delta f(x) = \sup_{x \in Q_x} \frac{1}{|Q_x|} \int_{Q_x} |f(y)| dy</math>

where ''Qx'' ranges over all [[dyadic cubes]] containing the point ''x''. Both of these operators satisfy the HL maximal inequality.

==References==
<references/>
*[[John B. Garnett]], ''Bounded Analytic Functions''. Springer-Verlag, 2006
*Antonios D. Melas, ''The best constant for the centered Hardy–Littlewood maximal inequality'', Annals of Mathematics, 157 (2003), 647–688
*Rami Shakarchi & [[Elias M. Stein]], ''Princeton Lectures in Analysis III: Real Analysis''. Princeton University Press, 2005
*Elias M. Stein, ''Maximal functions: spherical means'', Proc. Nat. Acad. Sci. U.S.A. '''73''' (1976), 2174–2175
*Elias M. Stein, ''Singular Integrals and Differentiability Properties of Functions''. Princeton University Press, 1971
*[[Gerald Teschl]], [http://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.html Topics in Real and Functional Analysis] (lecture notes)

{{DEFAULTSORT:Hardy-Littlewood Maximal Function}}
[[Category:Real analysis]]
[[Category:Harmonic analysis]]
[[Category:Types of functions]]

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en>Jaan Vajakas at 21:23, 22 May 2013