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  "body": "\n## Definition\n\nLet $M$ be a [[CR manifold|ZC78JKZX9S]] with CR structure $T^{0,1}M \\subseteq \\C TM$.\nFix a Hermitian metric on $T^{0,1}M$ and a smooth volume form on $M$.\nFor each integer $q \\ge 0$, let $\\mathcal{E}^{0,q}_b(M)$ be the space of smooth sections of $\\Lambda^q (T^{0,1}M)^*$, and let\n$$\\bar\\partial_b \\colon \\mathcal{E}^{0,q}_b(M) \\to \\mathcal{E}^{0,q+1}_b(M)$$\nbe the [[tangential Cauchy–Riemann operator|X7GP9BECD8]].\nThe metric and the volume form give an $L^2$ inner product on compactly supported forms in $\\mathcal{E}^{0,q}_b(M)$.\nLet $\\vartheta_b$ be the formal adjoint of $\\bar\\partial_b$ for these inner products, so that $\\ip{\\bar\\partial_b u}{v} = \\ip{u}{\\vartheta_b v}$ for compactly supported $u$ and $v$.\nThe *Kohn Laplacian* on $(0,q)$-forms is\n$$\\square_b = \\bar\\partial_b \\vartheta_b + \\vartheta_b \\bar\\partial_b \\colon \\mathcal{E}^{0,q}_b(M) \\to \\mathcal{E}^{0,q}_b(M).$$\n\n## Remarks\n\nThe adjoint $\\vartheta_b$ is often written $\\bar\\partial_b^*$, and $\\square_b$ is also called the Kohn–Rossi Laplacian or the boundary Laplacian.\nOn functions ($q = 0$) the first term is absent, and $\\square_b u = \\vartheta_b \\bar\\partial_b u$.\nExample: on the Heisenberg group $\\mathbb{H} = \\C \\times \\R$ with coordinates $(z, t)$, take $T^{0,1}\\mathbb{H}$ spanned by $\\bar Z = \\partial/\\partial \\bar z - i z\\, \\partial/\\partial t$, declare $\\bar Z$ to have length $1$, and use Lebesgue measure.\nThe vector field $\\bar Z$ has divergence $0$, so integration by parts shows that the formal adjoint of $\\bar Z$ is $-Z$, where $Z = \\partial/\\partial z + i \\bar z\\, \\partial/\\partial t$ is its complex conjugate.\nHence $\\square_b u = -Z \\bar Z u$ for functions $u$.\nNon-example: the operator $-\\bar Z Z$ is not the Kohn Laplacian on functions of $\\mathbb{H}$.\nIt differs from $-Z \\bar Z$ by the commutator $[Z, \\bar Z] = -2i\\, \\partial/\\partial t$, which is not zero.\nThe Kohn Laplacian is not elliptic, because its principal symbol vanishes on the nonzero real covectors that annihilate $T^{1,0}M \\oplus T^{0,1}M$.\n\n## Explanations\n\n### L1 human\n\nThe Kohn Laplacian is built from the derivatives along the complex directions tangent to a real hypersurface.\nLike the ordinary Laplacian, it is a sum of a first order operator followed by its adjoint, done in both orders.\nIt is the operator whose solutions describe the boundary values of holomorphic functions and forms.\n",
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