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)

: $f:mathbb{R}^{d}
ightarrow mathbb{C}$

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,

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

where

: $B_{r}(x)={yin mathbb{R}^{d}: ||y-x||$

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

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:

: $m_{d}{xinmathbb{R}^{d}: Mf(x)>lambda} {lambda}||f||_{L^{1}(mathbb{R}^{d})} .$

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

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

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 $1, we can remove the dependence of A_{p,d} on the dimension, that is, A_{p,d}=A_{p} for some constant A_{p}>0 only depending on the value p . It is unknown whether there is a weak bound that is independent of dimension.$

References

*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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