HARDY-LITTLEWOOD MAXIMAL FUNCTION

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In mathematics, the 'Hardy-Littlewood maximal operator' $M$ is a significant non-linear operator used in real analysis and harmonic analysis. It takes a function ''f'' (a complex-valued and locally integrable function)
:$f:mathbb\{R\}^\{d\}$
ightarrow mathbb{C}
and returns a second function
:$Mf\; ,$
that tells you, at each point $xin\; mathbb\{R\}^\{d\}$, how large the average value of $f$ can be on balls centered at that point. More precisely,
:
where
:
is the ball of radius $r$ centered at $x$), and $m\_\{d\}$ denotes the d-dimensional Lebesgue measure.
The averages are jointly continuous in ''x'' and ''r'', therefore the maximal function ''Mf'', being the supremum over ''r'' > 0, is measurable. It is not obvious that ''Mf'' is finite almost everywhere. This is a corollary of the 'Hardy-Littlewood maximal inequality'

Contents

Hardy-Littlewood maximal inequality

Proof

Applications

Discussion

References

Hardy-Littlewood maximal inequality

This theorem of G. H. Hardy and J. E. Littlewood states that $M$ is bounded as a sublinear operator from the ''L''^''p'' space
:$L^\{p\}(mathbb\{R\}^\{d\}),\; ;\; p\; >\; 1$
to itself. That is, if
:$fin\; L^\{p\}(mathbb\{R\}^\{d\}),$
then the maximal function ''Mf'' is weak ''L''¹ bounded and
:$Mfin\; L^\{p\}(mathbb\{R\}^\{d\}).$
More precisely, for all dimensions ''d'' ≥ 1 and 1 < ''p'' ≤ ∞, and all ''f'' ∈ ''L''¹('R'^''d''), there is a constant ''C_d'' > 0 such that for all ''λ'' > 0 , we have the ''weak type''-(1,1) bound:
:
This is the Hardy-Littlewood maximal inequality.
With the Hardy-Littlewood maximal inequality in hand, the following ''strong-type'' estimate is an immediate consequence of the Marcinkeiwicz interpolation theorem: there exists a constant ''A''_''p,d'' > 0 such that
:$||Mf||\_\{L^p(mathbb\{R\}^\{d\})\}leq\; A\_\{p,d\}||f||\_\{L^p(mathbb\{R\}^\{d\})\}.$

Proof

While there are several proofs of this theorem, a common one is outlined as follows: For $p=infty$, (see Lp space for definition of $L^\{infty\}$) the inequality is trivial (since the average of a function is no larger than its essential supremum). For 1 < ''p'' < ∞, one proves the weak bound using the Vitali covering lemma.

Applications

Some applications of the Hardy-Littlewood Maximal Inequality include proving the following results:

★ Lebesgue differentiation theorem

★ Rademacher differentiation theorem

★ Fatou's theorem on nontangential convergence.

Discussion

It is still unknown what the smallest constants $A\_\{p,d\}$ and $C\_\{d\}$ are in the above inequalities. However, a result of Elias Stein about spherical maximal functions can be used to show that, for

References

★ John B. Garnett, ''Bounded Analytic Functions''. Springer-Verlag, 2006

★ 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 & Guido Weiss, ''Singular Integrals and Differentiability Properties of Functions''. Princeton University Press, 1971