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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $\\abs{E}$ be the [[Lebesgue measure|2Z2QVWZ262]] of a set $E \\subseteq \\R^n$.\nFor a locally integrable function $f$ on $\\R^n$, let $Mf$ be its [[Hardy–Littlewood maximal function|VX1A56P5KQ]].\n\n1. For every $f \\in L^1(\\R^n)$ and every $\\lambda \u003e 0$,\n$$\\abs{\\set{x \\in \\R^n : Mf(x) \u003e \\lambda}} \\le \\frac{3^n}{\\lambda} \\norm{f}_{L^1}.$$\n2. For every $p$ with $1 \u003c p \\le \\infty$ there is a constant $C = C(n, p)$ such that $\\norm{Mf}_{L^p} \\le C \\norm{f}_{L^p}$ for every $f \\in L^p(\\R^n)$.\n\nHere $L^p(\\R^n)$ is the [[Lp space|SASP14MA1Z]] of Lebesgue measure.\n\n## Remarks\n\nPart 1 says that $M$ is of [[weak type|9534R6VMY0]] $(1, 1)$.\nIts proof uses the [[Vitali covering lemma|FCA0J1X07K]].\nEach point of a compact subset $K$ of $\\set{Mf \u003e \\lambda}$ is the centre of a ball $B$ with $\\int_B \\abs{f} \u003e \\lambda \\abs{B}$.\nThe covering lemma selects finitely many disjoint such balls $B_1, \\dots, B_m$ whose dilates $3B_k$ cover $K$, so $\\abs{K} \\le 3^n \\sum_k \\abs{B_k} \\le 3^n \\lambda^{-1} \\norm{f}_1$.\nFor $p = \\infty$, part 2 is the trivial bound $Mf \\le \\norm{f}_\\infty$, and for $1 \u003c p \u003c \\infty$ it follows from part 1 by the [[Marcinkiewicz interpolation theorem|R1PHMVBJ9J]].\n\nThe constant in part 2 grows like $1/(p - 1)$ as $p \\to 1$, and part 2 fails for $p = 1$.\nFor example, $f = \\mathbf{1}_{[0, 1]}$ on $\\R$ has $Mf(x) = 1/(2x)$ for $x \u003e 1$, which is not integrable.\nIn fact $Mf$ is integrable only when $f = 0$ almost everywhere.\n\n## Explanations\n\n### L1 human\n\nThe maximal function records, at each point, the largest average of $\\abs{f}$ over balls centred there.\nOne might fear that taking the largest of infinitely many averages makes the function huge.\nThe theorem says that this does not happen.\nFor $p \u003e 1$, the $L^p$ norm of $Mf$ is at most a constant times that of $f$.\nFor $p = 1$, a weaker bound holds on the size of the sets where $Mf$ is large.\n\n### L2 human\n\nThe inequality is the main tool for proving that averages converge almost everywhere.\nCombined with density of continuous functions it gives the Lebesgue differentiation theorem: the averages of $f$ over balls shrinking to $x$ tend to $f(x)$ for almost every $x$.\nMany operators in harmonic analysis, such as convolutions with radially decreasing integrable kernels, are bounded pointwise by a multiple of $Mf$, so they inherit its $L^p$ bounds.\nThe weighted version of the inequality leads to the Muckenhoupt $A_p$ condition.\n",
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