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  "body": "\n## Statement\n\nLet $n \\ge 1$ and $1 \\le p_0 \u003c \\infty$.\nFor a weight $w$ on $\\R^n$ and $1 \\le p \u003c \\infty$, write $\\norm{g}_{L^p(w)} = \\left( \\int_{\\R^n} \\abs{g}^p \\, w \\, dx \\right)^{1/p}$, and let $[w]_{A_p}$ be the constant of the [[Muckenhoupt class|T86RSJ8M86]] $A_p$.\nLet $\\mathcal{F}$ be a family of pairs $(f, g)$ of nonnegative measurable functions on $\\R^n$.\nAssume that there is an increasing function $N \\colon [1, \\infty) \\to [0, \\infty)$ such that\n$$\\norm{f}_{L^{p_0}(w)} \\le N([w]_{A_{p_0}}) \\, \\norm{g}_{L^{p_0}(w)}$$\nfor every $w \\in A_{p_0}$ and every $(f, g) \\in \\mathcal{F}$ with $\\norm{f}_{L^{p_0}(w)} \u003c \\infty$.\nThen let $1 \u003c p \u003c \\infty$ and $w \\in A_p$.\nThere is a constant $C$, depending only on $n$, $p$, $p_0$, $N$ and $[w]_{A_p}$, such that\n$$\\norm{f}_{L^p(w)} \\le C \\norm{g}_{L^p(w)}$$\nfor every $(f, g) \\in \\mathcal{F}$ with $\\norm{f}_{L^p(w)} \u003c \\infty$.\n\n## Remarks\n\nIn the typical use, the pairs are $(\\abs{Tf}, \\abs{f})$ for an operator $T$.\nAssume $T$ is bounded on $L^{p_0}(w)$ for all $w \\in A_{p_0}$, with a bound depending only on $[w]_{A_{p_0}}$.\nThen $T$ is bounded on $L^p(w)$ for all $1 \u003c p \u003c \\infty$ and all $w \\in A_p$.\nIn particular, taking $w = 1$, one exponent $p_0$ gives unweighted bounds for all $1 \u003c p \u003c \\infty$.\n\nThe proof uses the Rubio de Francia algorithm.\nLet $v$ be a weight and $q \u003e 1$ such that the [[maximal operator|VX1A56P5KQ]] $M$ is bounded on $L^q(v)$, with norm $\\norm{M}$.\nFor $h \\ge 0$ in $L^q(v)$, set $Rh = \\sum_{k \\ge 0} \\frac{M^k h}{(2\\norm{M})^k}$, where $M^k$ is the $k$-th iterate.\nThen $h \\le Rh$, $\\norm{Rh}_{L^q(v)} \\le 2 \\norm{h}_{L^q(v)}$, and $M(Rh) \\le 2 \\norm{M} \\, Rh$, so $Rh$ is an $A_1$ weight.\nThese $A_1$ weights are combined into an $A_{p_0}$ weight to which the hypothesis applies, and [[Muckenhoupt's theorem|S9BKNBRV4H]] controls the constants.\n\n## Explanations\n\n### L1 human\n\nProving an inequality for one exponent $p$ is often much easier than for all of them, for example $p = 2$ where Fourier methods work.\nExtrapolation says that if the inequality holds for one exponent and for all Muckenhoupt weights at that exponent, then it holds for all exponents $1 \u003c p \u003c \\infty$ and all corresponding weights.\nSo weights act as a bridge between different exponents.\n\n### L2 human\n\nThe theorem is a main reason to prove weighted inequalities.\nA single weighted $L^2$ bound for a Calderón–Zygmund operator, a commutator or a square function yields all the $L^p$ bounds, weighted and unweighted.\nIt also gives vector-valued inequalities almost for free, since a sum $\\left( \\sum_j \\abs{T f_j}^q \\right)^{1/q}$ can be treated by applying the pair formulation to suitable functions.\nIt relies on the theory of $A_p$ weights, in particular on Muckenhoupt's theorem and on the factorization of $A_p$ weights as $w = w_1 w_2^{1 - p}$ with $w_1, w_2 \\in A_1$.\n",
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    "name": "Rubio de Francia extrapolation theorem",
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