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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) ## Hardy-Littlewood maximal inequality

## 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:## 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 $1math>,\; we\; can\; remove\; the\; dependence\; of$ A\_\{p,d\}$on\; the\; dimension,\; that\; is,$ A\_\{p,d\}=A\_\{p\}$for\; some\; constant$ A\_\{p\}0$only\; depending\; on\; the\; value$ p$.\; It\; is\; unknown\; whether\; there\; is\; a\; weak\; bound\; that\; is\; independent\; of\; dimension.$## References

- $f:mathbb\{R\}^\{d\}rightarrow\; mathbb\{C\}$

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,

- $Mf(x)=sup\_\{r>0\}frac\{1\}\{m\_d(B\_\{r\}(x))\}int\_\{B\_\{r\}(x)\}\; |f(y)|\; dm\_\{d\}(y)$

where

- $B\_\{r\}(x)=\{yin\; mathbb\{R\}^\{d\}:\; ||y-x||\}\; math>$

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

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:

- $m\_\{d\}\{xinmathbb\{R\}^\{d\}:\; Mf(x)>lambda\}\{c\_\{d\}\}\{lambda\}||f||\_\{l^\{1\}(mathbb\{r\}^\{d\})\}\; .\; math>$

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 A_{p,d} > 0 such that

- $||Mf||\_\{L^p(mathbb\{R\}^\{d\})\}leq\; A\_\{p,d\}||f||\_\{L^p(mathbb\{R\}^\{d\})\}.$

- Lebesgue differentiation theorem
- Rademacher differentiation theorem
- Fatou's theorem on nontangential convergence.

- 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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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday May 10, 2008 at 13:12:52 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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