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  "body": "\n## Summary\n\nA hypoelliptic operator is a linear differential operator $P$ with smooth coefficients such that every distribution $u$ with $Pu$ smooth is itself smooth.\nSo $P$ cannot create singularities in its solutions that are not already in the right-hand side.\n[[Elliptic operators|XY8GEQSWPR]] with smooth coefficients are hypoelliptic, and so is the heat operator, which is not elliptic.\nHörmander characterized hypoelliptic operators with constant coefficients and gave a wide class of hypoelliptic sums of squares of vector fields.\n\n## Explanations\n\n### L1 human\n\nSome equations smooth things out: any solution of $\\Delta u = 0$, even a very rough one at first sight, turns out to be infinitely differentiable.\nHypoelliptic operators are those with this smoothing property, whatever smooth right-hand side one puts in.\nThe heat operator $\\partial_t - \\Delta$ is one: temperature profiles are smooth even if the operator only involves one time derivative.\nThe wave operator is not: a sharp wave front $u(x, t) = H(x - t)$, with $H$ the jump function, solves the one-dimensional wave equation and keeps its jump forever.\n",
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