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In
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 ...
, the Hardy–Littlewood maximal operator ''M'' is a significant non-linear
operator Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
used in
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...
and
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 ...
.


Definition

The operator takes a
locally integrable In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions li ...
function f: \R^d \to \mathbb C and returns another function Mf : \R^d \to , \infty/math>, where Mf (x) is the supremum of the average of f among all possible balls centered on x. Formally, : Mf(x)=\sup_ \frac\int_ , f(y), \, dy , where , ''E'', denotes the ''d''-dimensional Lebesgue measure of a subset ''E'' ⊂ R''d'', and B(x,\,r) is the ball of radius, r>0, centered at the point x\in\mathbb^d. Since f is locally integrable, the averages are jointly
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
in ''x'' and ''r'', so the maximal function ''Mf'', being the supremum over ''r'' > 0, is
measurable In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts hav ...
. A nontrivial corollary of the Hardy–Littlewood maximal inequality states that Mf is finite
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
for functions in L^1 .


Hardy–Littlewood maximal inequality

This theorem of
G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
and J. E. Littlewood states that ''M'' is bounded as a sublinear operator from ''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 denote the set . Now we have:
Theorem (Weak Type Estimate). For ''d'' ≥ 1, there is a constant ''Cd'' > 0 such that for all λ > 0 and ''f'' ∈ ''L''1(R''d''), we have: :\left , \ \right , < \frac \Vert f\Vert_.
With the Hardy–Littlewood maximal inequality in hand, the following ''strong-type'' estimate is an immediate consequence of the Marcinkiewicz interpolation theorem:
Theorem (Strong Type Estimate). For ''d'' ≥ 1, 1 < ''p'' ≤ ∞, and ''f'' ∈ ''Lp''(R''d''), there is a constant ''Cp,d'' > 0 such that : \Vert Mf\Vert_\leq C_\Vert f\Vert_.
In the strong type estimate the best bounds for ''Cp,d'' are unknown. However subsequently Elias M. Stein used the Calderón-Zygmund method of rotations to prove the following:
Theorem (Dimension Independence). For 1 < ''p'' ≤ ∞ one can pick ''Cp,d'' = ''Cp'' independent of ''d''.


Proof

While there are several proofs of this theorem, a common one is given below, that uses the following version of the Vitali covering lemma to prove the weak-type estimate. (See the article for the proof of the lemma.) Note that the constant C=5^d in the proof can be improved to 3^d by using the inner regularity of the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
, and the finite version of the Vitali covering lemma. See the
Discussion section A recitation in a general sense is the act of reciting from memory, or a formal reading of verse or other writing before an audience. Public recitation is the act of reciting a work of writing before an audience. Academic recitation In a ...
below for more about optimizing the constant.


Applications

Some applications of the Hardy–Littlewood Maximal Inequality include proving the following results: * Lebesgue differentiation theorem * Rademacher differentiation theorem *
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 ...
on nontangential convergence. *
Fractional integration theorem A fraction is one or more equal parts of something. Fraction may also refer to: * Fraction (chemistry), a quantity of a substance collected by fractionation * Fraction (floating point number), an (ambiguous) term sometimes used to specify a part ...
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 :\Omega f (x) = \limsup_ f_r(x) - \liminf_ f_r(x) where :f_r(x) = \frac \int_ f(y) dy. 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 :\Omega f \le \Omega g + \Omega h = \Omega g by continuity. Now, Ω''g'' ≤ 2''Mg'' and so, by the theorem, we have: :\left , \ \right , \le \frac \, g\, _1 Now, we can let \, g\, _1 \to 0 and conclude Ω''f'' = 0 almost everywhere; that is, \lim_ f_r(x) exists for almost all ''x''. It remains to show the limit actually equals ''f''(''x''). But this is easy: it is known that \, f_r - f\, _1 \to 0 ( approximation of the identity) and thus there is a subsequence f_ \to f 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) : f^*(x) = \sup_ \frac \int_ , f(y), dy where the balls ''Bx'' are required to merely contain x, rather than be centered at x. There is also the ''dyadic'' HL maximal operator :M_\Delta f(x) = \sup_ \frac \int_ , f(y), dy where ''Qx'' ranges over all
dyadic cubes Dyadic describes the interaction between two things, and may refer to: *Dyad (sociology), interaction between a pair of individuals **The dyadic variation of democratic peace theory *Dyadic counterpoint, the voice-against-voice conception of polyp ...
containing the point ''x''. Both of these operators satisfy the HL maximal inequality.


See also

* Rising sun lemma


References

* John B. Garnett, ''Bounded Analytic Functions''. Springer-Verlag, 2006 *G. H. Hardy and J. E. Littlewood. ''A maximal theorem with function-theoretic applications''. Acta Math. 54, 81–116 (1930)

*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. Natl. 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 Gerald Teschl (born 12 May 1970 in Graz) is an Austrian mathematical physicist and professor of mathematics. He works in the area of mathematical physics; in particular direct and inverse spectral theory with application to completely integrable ...

Topics in Real and Functional Analysis
(lecture notes) {{DEFAULTSORT:Hardy-Littlewood Maximal Function Real analysis Harmonic analysis Types of functions