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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $\\mathcal{D}$ be the space of smooth functions with compact support on $\\R^n$, and let $\\mathcal{D}'$ be its dual, the space of distributions.\nLet $T \\colon \\mathcal{D} \\to \\mathcal{D}'$ be a continuous linear operator with transpose $T^t$, defined by $\\langle T^t f, g \\rangle = \\langle Tg, f \\rangle$ for $f, g \\in \\mathcal{D}$.\nAssume that $T$ is associated with a [[Calderón–Zygmund kernel|78N6AXP9VV]] $K$: for $f, g \\in \\mathcal{D}$ with disjoint supports,\n$$\\langle Tf, g \\rangle = \\int\\!\\!\\int K(x, y) f(y) g(x) \\, dy \\, dx.$$\nThen $T$ extends to a bounded operator on $L^2(\\R^n)$ if and only if the following three conditions hold:\n\n1. $T(1)$ lies in the space $\\mathrm{BMO}$ of functions of [[bounded mean oscillation|QJYTXFT8Y4]];\n2. $T^t(1)$ lies in $\\mathrm{BMO}$;\n3. $T$ has the weak boundedness property.\n\nHere $T(1)$ and $T^t(1)$ are defined as elements of $\\mathrm{BMO}$ modulo constants, using the kernel bounds.\nThe weak boundedness property means that there are constants $C$ and $N$ with the following property.\nLet $B$ be a ball of radius $R$, and let $\\phi, \\psi \\in \\mathcal{D}$ be supported in $B$ with $\\norm{\\partial^\\alpha\\phi}_\\infty, \\norm{\\partial^\\alpha\\psi}_\\infty \\le R^{-\\abs{\\alpha}}$ for $\\abs{\\alpha} \\le N$.\nThen $\\abs{\\langle T\\phi, \\psi \\rangle} \\le C R^n$.\n\n## Remarks\n\nGuy David and Jean-Lin Journé proved the theorem in 1984.\nThe three conditions are necessary.\nAn $L^2$-bounded operator with a Calderón–Zygmund kernel maps $L^\\infty$ into $\\mathrm{BMO}$, and so does its transpose.\nThe weak boundedness property follows from the $L^2$ bound and the Cauchy–Schwarz inequality.\nFor sufficiency, one subtracts paraproducts with symbols $T(1)$ and $T^t(1)$, which are $L^2$ bounded, and reduces to the case $T(1) = T^t(1) = 0$.\nThat case is handled by an almost orthogonality argument, such as the Cotlar–Stein lemma, applied to a Littlewood–Paley decomposition of $T$.\n\nExample: for an antisymmetric kernel, $K(x, y) = -K(y, x)$, with $T$ defined as a principal value, the weak boundedness property holds automatically.\nThen the theorem reduces to the single condition $T(1) \\in \\mathrm{BMO}$.\nA model case is the Hilbert transform, for which $T(1) = 0$.\n\nWithout conditions 1 and 2 the conclusion fails: the kernel bounds alone do not give $L^2$ boundedness, since they carry no information on the cancellation of $T$.\n\n## Explanations\n\n### L1 human\n\nFor a singular integral operator, the hard part is to show that it is bounded on $L^2$; the rest of the theory then gives $L^p$ bounds.\nThe $T(1)$ theorem reduces this hard question to testing the operator on the constant function $1$.\nIf $T$ and its transpose send $1$ to functions of bounded mean oscillation, and $T$ is mild on bump functions, then $T$ is bounded on $L^2$.\n\n### L2 human\n\nThe theorem turns $L^2$ bounds for non-convolution operators, which cannot be checked with the Fourier transform, into simple tests.\nIt gives a short proof of the $L^2$ boundedness of the Cauchy integral on Lipschitz curves and of the Calderón commutators.\nIts output, an $L^2$-bounded operator with a Calderón–Zygmund kernel, is a [[Calderón–Zygmund operator|3SCXCW4AH3]], hence bounded on $L^p$ for $1 \u003c p \u003c \\infty$.\nThere are versions with $T(b)$ in place of $T(1)$, for suitable accretive functions $b$.\n",
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