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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be an open set and let $P = \\sum_{\\abs{\\alpha} \\le m} a_\\alpha(x) \\partial^\\alpha$ be a linear differential operator on $\\Omega$ whose coefficients $a_\\alpha$ are [[smooth functions|0V33BNKZ9S]].\nA distribution on an open set $U$ is a continuous linear functional on the space of test functions on $U$.\nThe operator $P$ is *hypoelliptic* if the following holds for every open set $U \\subseteq \\Omega$ and every distribution $u$ on $U$.\nIf $Pu$ is a smooth function on $U$, then $u$ is a smooth function on $U$.\n\n## Remarks\n\nEquivalently, $\\operatorname{sing\\,supp} u = \\operatorname{sing\\,supp} Pu$ for every distribution $u$, where the singular support is the complement of the largest open set on which the distribution is smooth.\nExample: every [[elliptic operator|XY8GEQSWPR]] with smooth coefficients is hypoelliptic; this is elliptic regularity.\nThe heat operator $\\partial_t - \\Delta$ on $\\R^{n+1}$ is hypoelliptic but not elliptic.\nNon-example: the wave operator $\\partial_t^2 - \\partial_x^2$ on $\\R^2$ is not hypoelliptic.\nFor the Heaviside function $H$, the distribution $u(x, t) = H(x - t)$ satisfies $\\partial_t^2 u - \\partial_x^2 u = H''(x - t) - H''(x - t) = 0$, which is smooth, but $u$ is not smooth.\n",
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