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  "body": "\n## Summary\n\nThe Kohn Laplacian is the second order operator $\\square_b = \\bar\\partial_b \\bar\\partial_b^* + \\bar\\partial_b^* \\bar\\partial_b$ on a [[CR manifold|ZC78JKZX9S]], built from the [[tangential Cauchy–Riemann operator|X7GP9BECD8]] $\\bar\\partial_b$ and its adjoint.\nIt plays on the boundary of a domain in $\\C^n$ the role that the complex Laplacian $\\bar\\partial \\bar\\partial^* + \\bar\\partial^* \\bar\\partial$ of the [[dbar-Neumann problem|Q4P4R2GDNG]] plays inside.\nIt is not elliptic, but on strongly pseudoconvex CR manifolds it is hypoelliptic in suitable degrees.\nIts study gives the solution of the tangential Cauchy–Riemann equations and the [[Kohn–Rossi cohomology|VC48WDWPV1]].\n\n## Explanations\n\n### L1 human\n\nOn a real hypersurface in $\\C^n$, such as a sphere, only some of the complex directions are tangent to the surface.\nThe tangential Cauchy–Riemann operator differentiates only in these directions.\nThe Kohn Laplacian combines this operator with its adjoint, in the same way as the usual Laplacian combines the gradient with the divergence.\nOn the Heisenberg group $\\C \\times \\R$, which is a model of the boundary of a ball, it acts on functions as $-Z \\bar Z$ for the vector fields $\\bar Z = \\partial/\\partial \\bar z - i z\\, \\partial/\\partial t$ and $Z = \\partial/\\partial z + i \\bar z\\, \\partial/\\partial t$.\nIt differentiates twice in the $z$ directions but only indirectly in the $t$ direction, which is why it is not elliptic.\n",
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