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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $w$ be a weight on $\\R^n$, and let $M$ be the [[Hardy–Littlewood maximal function|VX1A56P5KQ]] operator.\nFor $1 \\le p \u003c \\infty$ write $L^p(w)$ for the [[Lp space|SASP14MA1Z]] of the measure $w(x) \\, dx$, and $w(E) = \\int_E w \\, dx$.\n\n1. For $1 \u003c p \u003c \\infty$, there is a constant $C$ with $\\int_{\\R^n} (Mf)^p \\, w \\, dx \\le C \\int_{\\R^n} \\abs{f}^p \\, w \\, dx$ for all $f \\in L^p(w)$ if and only if $w$ is in the [[Muckenhoupt class|T86RSJ8M86]] $A_p$.\n2. There is a constant $C$ with $w(\\set{Mf \u003e \\lambda}) \\le \\frac{C}{\\lambda} \\int_{\\R^n} \\abs{f} \\, w \\, dx$ for all $f \\in L^1(w)$ and $\\lambda \u003e 0$ if and only if $w \\in A_1$.\n\n## Remarks\n\nNecessity in part 1 comes from testing on one cube.\nLet $\\sigma = w^{-1/(p - 1)}$ and $f = \\sigma \\mathbf{1}_Q$.\nOn $Q$ one has $Mf \\ge \\frac{1}{\\abs{Q}} \\int_Q \\sigma$, and inserting this in the inequality gives the $A_p$ bound for $Q$, after a cut-off to make $f \\in L^p(w)$.\nFor sufficiency, one first shows the [[weak-type|9534R6VMY0]] bound on $L^p(w)$ from the $A_p$ condition and the covering argument.\nThen one uses that an $A_p$ weight is also in $A_{p - \\varepsilon}$ for some $\\varepsilon \u003e 0$, which follows from a reverse Hölder inequality, and interpolates with the [[Marcinkiewicz interpolation theorem|R1PHMVBJ9J]].\n\nExample: $w(x) = \\abs{x}$ on $\\R$ is not in $A_2$, and indeed the weighted inequality fails for $p = 2$.\nFor $0 \u003c \\delta \u003c 1$ let $f = \\abs{x}^{-1} \\mathbf{1}_{[\\delta, 1]}$; then $\\int f^2 w = \\log(1/\\delta)$.\nFor $x \\in [-1, 1]$ the interval of radius $2$ centred at $x$ contains $[\\delta, 1]$, so $Mf(x) \\ge \\frac{1}{4} \\log(1/\\delta)$.\nHence $\\int (Mf)^2 w \\ge \\frac{1}{16} \\log^2(1/\\delta)$, and the ratio of the two sides tends to infinity as $\\delta \\to 0$.\n\n## Explanations\n\n### L1 human\n\nThe maximal function is bounded on ordinary $L^p$ for $p \u003e 1$.\nThe theorem says for which weights this stays true when the $L^p$ norm is weighted.\nThe answer is a simple test on cubes, the $A_p$ condition.\n\n### L2 human\n\nMuckenhoupt proved the theorem in 1972.\nShortly after, Hunt, Muckenhoupt and Wheeden showed that the same $A_p$ condition characterizes the weights for which the [[Hilbert transform|NSCEBXM7ZM]] is bounded on $L^p(w)$, $1 \u003c p \u003c \\infty$.\nCoifman and Fefferman extended the sufficiency to general Calderón–Zygmund operators.\nThe theorem is the starting point of weighted theory, including extrapolation and the sharp dependence of operator norms on $[w]_{A_p}$.\n",
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