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  "body": "\n## Definition\n\nLet $n \\ge 1$.\nA *weight* on $\\R^n$ is a locally [[integrable|X7GTF3D98F]] function $w \\colon \\R^n \\to [0, \\infty)$ with $w \u003e 0$ [[almost everywhere|RARTMA0ZEC]].\nFor a cube $Q \\subseteq \\R^n$ with sides parallel to the axes, let $\\abs{Q}$ be its [[Lebesgue measure|2Z2QVWZ262]].\n\n1. For $1 \u003c p \u003c \\infty$, the weight $w$ is in the *Muckenhoupt class* $A_p$ if\n$$[w]_{A_p} = \\sup_Q \\left( \\frac{1}{\\abs{Q}} \\int_Q w \\, dx \\right) \\left( \\frac{1}{\\abs{Q}} \\int_Q w^{-1/(p - 1)} \\, dx \\right)^{p - 1} \u003c \\infty,$$\nwhere the supremum is over all such cubes $Q$.\n2. The weight $w$ is in $A_1$ if there is a constant $C$ with $Mw(x) \\le C w(x)$ for almost every $x$, where $M$ is the [[Hardy–Littlewood maximal function|VX1A56P5KQ]].\n\nA weight in some class $A_p$ is called a *Muckenhoupt weight*.\n\n## Remarks\n\nBy Hölder's inequality $[w]_{A_p} \\ge 1$, and $A_1 \\subseteq A_p \\subseteq A_q$ for $1 \u003c p \u003c q$.\nThe union $A_\\infty = \\bigcup_{p \\ge 1} A_p$ has several equivalent descriptions, for example through a reverse Hölder inequality.\nFor $p = 2$ the condition reads $\\sup_Q \\left( \\frac{1}{\\abs{Q}}\\int_Q w \\right) \\left( \\frac{1}{\\abs{Q}}\\int_Q w^{-1} \\right) \u003c \\infty$.\n\nExample: $n = 1$, $p = 2$, $w(x) = \\abs{x}^a$ with $-1 \u003c a \u003c 1$.\nOn $Q = [0, L]$ the two averages are $L^a/(1 + a)$ and $L^{-a}/(1 - a)$, whose product is $1/(1 - a^2)$ for every $L$.\nIntervals symmetric about $0$ give the same averages.\nOn intervals far from $0$ compared with their length, $w$ is almost constant, so the product is close to $1$.\nThe remaining intervals lie in a symmetric interval of comparable length, and this case analysis shows $w \\in A_2$.\nMore generally $\\abs{x}^a \\in A_p(\\R^n)$ exactly when $-n \u003c a \u003c n(p - 1)$.\n\nNon-example: $n = 1$, $p = 2$, $w(x) = \\abs{x}$.\nIt is a weight, but on $Q = [0, 1]$ the average of $w^{-1} = 1/x$ is infinite, so $w \\notin A_2$.\n",
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