Definitions

# 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. It takes a function f (a complex-valued and locally integrable function)

$f:mathbb\left\{R\right\}^\left\{d\right\}rightarrow mathbb\left\{C\right\}$

and returns a second function

$Mf ,$

that tells you, at each point $xin mathbb\left\{R\right\}^\left\{d\right\}$, how large the average value of $f$ can be on balls centered at that point. More precisely,

$Mf\left(x\right)=sup_\left\{r>0\right\}frac\left\{1\right\}\left\{m_d\left(B_\left\{r\right\}\left(x\right)\right)\right\}int_\left\{B_\left\{r\right\}\left(x\right)\right\} |f\left(y\right)| dm_\left\{d\right\}\left(y\right)$

where

is the ball of radius $r$ centered at $x$), and $m_\left\{d\right\}$ 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

## 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 Lp space

$L^\left\{p\right\}\left(mathbb\left\{R\right\}^\left\{d\right\}\right), ; p > 1$

to itself. That is, if

$fin L^\left\{p\right\}\left(mathbb\left\{R\right\}^\left\{d\right\}\right),$

then the maximal function Mf is weak L1 bounded and

$Mfin L^\left\{p\right\}\left(mathbb\left\{R\right\}^\left\{d\right\}\right).$

More precisely, for all dimensions d ≥ 1 and 1 < p ≤ ∞, and all fL1(Rd), there is a constant Cd > 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 Marcinkiewicz interpolation theorem: there exists a constant Ap,d > 0 such that

$||Mf||_\left\{L^p\left(mathbb\left\{R\right\}^\left\{d\right\}\right)\right\}leq A_\left\{p,d\right\}||f||_\left\{L^p\left(mathbb\left\{R\right\}^\left\{d\right\}\right)\right\}.$

## 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 $L^\left\{infty\right\}$) 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:

## Discussion

It is still unknown what the smallest constants $A_\left\{p,d\right\}$ and $C_\left\{d\right\}$ 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
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