Hardy-Littlewood maximal function ($Mf$)

The centered Hardy-Littlewood maximal function on the real line is

If $k(x)=\varphi (|x|)\ge 0$ is decreasing in radius and has integral one, then $|k_t\ast f(x)|\le Mf(x)$, where $k_t(x)=t^{-d}k(x/t)$ and $M$ is the centered Hardy-Littlewood maximal function of $|f|dx$. The layer cake representation expresses $k_t$ as a positive mixture of ball indicators; each corresponding integral is bounded by its ball volume times $Mf(x)$. Integrating these volumes gives exactly one. This applies to the Poisson kernel for the upper half-space and supplies a weak type (1,1) domination for its boundary convergence.

The uncentered maximal function takes the supremum over all open Euclidean balls containing $x$, not only balls centred at $x$. It is an extended nonnegative lower semicontinuous function: each strict superlevel set is the union of the balls whose average density exceeds that level. For finite positive measures it satisfies the uncentered maximal weak-type inequality.

For $r>0$, the forward average is $r^{-1}{\int }_x^{x+r}f(t)dt$ and the backward average is $r^{-1}{\int }_{x-r}^{x}f(t)dt$, both with positive interval length in the denominator. They are linear operators. Their absolute values are bounded by the uncentered maximal function: enlarge the interval to put $x$ in its interior and pass to the limit. The same boundary-enlargement argument works regardless of the value of the chosen representative at $x$.

For a finite positive measure on Euclidean space, cover a compact subset of the maximal superlevel set by finitely many high-density balls. Wiener covering lemma selects disjoint balls whose triples cover it. Their total measure mass is at most the mass of the original measure. Inner regularity of Lebesgue measure yields the displayed bound. Applied to approximation errors, it proves the Lebesgue differentiation theorem.

For a positive Borel measure on ${\mathbb{R}}^d$, let $E_{\alpha }=\{M_u\nu >\alpha \}$, using open balls in the uncentered maximal function of a finite measure. Every ball witnessing the inequality lies entirely in $E_{\alpha }$. Cover a compact subset by finitely many witnessing balls and select disjoint balls greedily by decreasing radius. Their threefold dilations cover the compact set. Their total measure is at most $\nu (E_{\alpha })$, giving the bound by inner regularity of Lebesgue measure. For a signed or complex measure, apply this to its total variation measure.

The maximal averaging operator has weak type $(1,1)$, controlling the measure of points where local averages are large.

For $1<p\le \infty$, the Hardy-Littlewood maximal function is bounded on $L^p({\mathbb{R}}^n)$. A direct proof for finite $p$ applies the weak $(1,1)$ estimate to $f{1}_{\{|f|>s/2\}}$, integrates the resulting distribution-function bound against $p{s}^{p-1}$, and uses Tonelli theorem.