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  "body": "\n## Statement\n\nLet $n \\ge 1$ and $1 \u003c p \u003c \\infty$.\nFor a locally integrable function $b$ on $\\R^n$ and an operator $T$, let $[b, T]$ be the commutator $[b, T] f = b \\, Tf - T(bf)$, defined for smooth $f$ with compact support.\n\nLet $T$ be a [[singular integral operator|HTMCEARGDD]] with kernel $\\Omega(x/\\abs{x})/\\abs{x}^n$.\nHere $\\Omega$ is infinitely differentiable on the unit sphere and has mean zero.\nLet $R_1, \\dots, R_n$ be the [[Riesz transforms|FRNDFJ9WQ5]], and write $\\norm{\\cdot}$ for the operator norm on the [[Lp space|SASP14MA1Z]] $L^p(\\R^n)$.\n\n1. If $b$ has [[bounded mean oscillation|QJYTXFT8Y4]], then $[b, T]$ extends to a bounded operator on $L^p(\\R^n)$.\nMoreover $\\norm{[b, T]} \\le C \\norm{b}_{\\mathrm{BMO}}$, where $C$ depends only on $n$, $p$ and $\\Omega$.\n2. If $[b, R_j]$ is bounded on $L^p(\\R^n)$ for every $j = 1, \\dots, n$, then $b$ is in BMO.\nMoreover $\\norm{b}_{\\mathrm{BMO}} \\le C \\sum_{j=1}^n \\norm{[b, R_j]}$, where $C$ depends only on $n$ and $p$.\n\n## Remarks\n\nOne proof of part 1 uses a conjugation trick.\nFor complex $z$, the operator $T_z f = e^{zb} T(e^{-zb} f)$ satisfies $\\frac{d}{dz} T_z \\big\\vert_{z = 0} = [b, T]$.\nBy the John–Nirenberg inequality, $e^{p \\, \\mathrm{Re}(z) b}$ is a Muckenhoupt $A_p$ weight when $\\abs{z} \\norm{b}_{\\mathrm{BMO}}$ is small.\nSo, by weighted theory, $T_z$ is bounded on $L^p$ uniformly for small $\\abs{z}$.\nThe Cauchy integral formula for the derivative then gives the bound on $[b, T]$.\n\nFor $n = 1$ the only Riesz transform is the [[Hilbert transform|NSCEBXM7ZM]] $H$, and the theorem says that $[b, H]$ is bounded on $L^p(\\R)$ exactly when $b \\in \\mathrm{BMO}(\\R)$.\nExample: $b(x) = \\log \\abs{x}$ is unbounded, so the multiplication operator by $b$ is unbounded on $L^p$, yet $[b, H]$ is bounded because $b \\in \\mathrm{BMO}$.\n\n## Explanations\n\n### L1 human\n\nA commutator $[b, T]$ measures how much the order of two operations matters: first apply $T$ and then multiply by $b$, or first multiply by $b$ and then apply $T$.\nFor a singular integral $T$, the theorem says that this difference is a bounded operator exactly when $b$ has bounded mean oscillation.\nSo BMO is precisely the class of multipliers that almost commute with singular integrals.\n\n### L2 human\n\nThe theorem gives a new characterization of BMO through operators, complementing its definition by mean oscillations and Fefferman's duality with $H^1$.\nBy duality it is equivalent to a factorization theorem for $H^1$: every $f \\in H^1(\\R^n)$ can be written as $\\sum_k \\sum_j \\left( g_{k,j} R_j h_{k,j} + h_{k,j} R_j g_{k,j} \\right)$ with $\\sum_k \\sum_j \\norm{g_{k,j}}_{p'} \\norm{h_{k,j}}_p \u003c \\infty$.\nCommutators with BMO functions appear in the study of div-curl lemmas in compensated compactness and of elliptic equations with VMO coefficients.\nThe commutator $[b, T]$ is compact on $L^p$ exactly when $b$ is in the closure of smooth compactly supported functions in BMO, a result of Uchiyama.\n",
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