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 ƒ (a complex-valued and locally integrable function)

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

and returns a second function

$Mf\,$

that, at each point x ∈ R^d, gives the maximum average value that ƒ can have 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

$B_{r}(x)=\{y\in \mathbb {R} ^{d}:\Vert y-x\Vert <r\}\,$

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

$L^{p}(\mathbb {R} ^{d}),\quad p>1$

to itself. That is, if

$f\in L^{p}(\mathbb {R} ^{d}),$

then the maximal function Mf is weak L¹ bounded and

$Mf\in L^{p}(\mathbb {R} ^{d}).\,$

More precisely, for all dimensions d ≥ 1 and 1 < p ≤ ∞, and all ƒ ∈ L¹(R^d), there is a constant C_d > 0 such that for all λ > 0, we have the weak type-(1,1) bound:

$m_{d}\{x\in \mathbb {R} ^{d}:Mf(x)>\lambda \}<{\frac {C_{d}}{\lambda }}\Vert f\Vert _{L^{1}(\mathbb {R} ^{d})}.$

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

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

Subsequent work by Elias Stein used the Calderón-Zygmund method of rotations to show that one could pick A_p,d = A_p independent of d. The best bounds for A_p,d are unknown.

Proof

While there are several proofs of this theorem, a common one is outlined as follows: For p = ∞, (see Lp space for definition of L^{∞) 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 < p< ∞, 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
• Antonios D. Melas, The best constant for the centered Hardy–Littlewood maximal inequality, Annals of Mathematics, 157 (2003), 647–688