Maximal functions

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}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)


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 Lp 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 L1 bounded and

Mfin L^{p}(mathbb{R}^{d}).

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:

m_{d}{xinmathbb{R}^{d}: Mf(x)>lambda}

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})}.


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.


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


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.


  • 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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