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}\rightarrow \mathbb{C}

and returns a second function

Mf \,

that tells you, at each point x\in \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)=\{y\in \mathbb{R}^{d}: ||y-x||<r\}

is the ball of radius r centered at x), and md 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

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[edit] 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^{p}(\mathbb{R}^{d}), \; 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 f ∈ L1(Rd), there is a constant Cd > 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}||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 Ap,d > 0 such that

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

[edit] 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.

[edit] Applications

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

[edit] 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<\infty, 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.

[edit] 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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