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  "body": "\n## Definition\n\nLet $\\Omega$ be an open subset of $\\R^n$ and let $u$ be a distribution on $\\Omega$.\nA *cone* is a set $\\Gamma \\subseteq \\R^n \\setminus \\set{0}$ such that $t \\xi \\in \\Gamma$ for all $\\xi \\in \\Gamma$ and all $t \u003e 0$.\nA point $(x_0, \\xi_0) \\in \\Omega \\times (\\R^n \\setminus \\set{0})$ is *microlocally regular* for $u$ if there exist:\n\n1. a test function $\\varphi \\in C_c^\\infty(\\Omega)$ with $\\varphi(x_0) \\ne 0$,\n2. an open cone $\\Gamma$ with $\\xi_0 \\in \\Gamma$,\n3. for each integer $N \\ge 0$ a constant $C_N$,\n\nsuch that the [[Fourier transform|DPVM34ADZV]] of the compactly supported distribution $\\varphi u$ satisfies $\\abs{\\widehat{\\varphi u}(\\xi)} \\le C_N (1 + \\abs{\\xi})^{-N}$ for all $\\xi \\in \\Gamma$ and all $N$.\nThe *wave front set* $\\operatorname{WF}(u)$ is the set of points of $\\Omega \\times (\\R^n \\setminus \\set{0})$ that are not microlocally regular for $u$.\n\n## Remarks\n\nSince $\\varphi u$ has compact support, $\\widehat{\\varphi u}$ is a [[smooth function|0V33BNKZ9S]], so the decay condition makes sense pointwise.\nBy construction $\\operatorname{WF}(u)$ is closed, and it is conic: $(x, \\xi) \\in \\operatorname{WF}(u)$ implies $(x, t\\xi) \\in \\operatorname{WF}(u)$ for $t \u003e 0$.\nIts projection to $\\Omega$ is the singular support of $u$, the complement of the largest open set on which $u$ is smooth.\nIn particular $\\operatorname{WF}(u) = \\emptyset$ if and only if $u$ is smooth.\n\nExample: for the [[Dirac delta|0KPHENZ5S7]] $\\delta$ at $0$ in $\\R^n$, one has $\\operatorname{WF}(\\delta) = \\set{0} \\times (\\R^n \\setminus \\set{0})$.\nCheck: at a point $x_0 \\ne 0$ one may take $\\varphi$ vanishing near $0$, so $\\varphi \\delta = 0$.\nAt $x_0 = 0$, $\\widehat{\\varphi \\delta}(\\xi) = \\varphi(0)$ is a nonzero constant, which decays in no direction.\n\nNon-example: let $u(x) = H(x_1)$ on $\\R^2$, where $H$ is the [[Heaviside step function|09C4JP7Y0M]], and look at the point $(0, (0, 1))$.\nTake $\\varphi(x) = \\chi(x_1) \\chi(x_2)$ with $\\chi \\in C_c^\\infty(\\R)$ and $\\chi(0) \\ne 0$.\nThen $\\widehat{\\varphi u}(\\xi) = a(\\xi_1)\\, \\hat\\chi(\\xi_2)$, where $a$ is the Fourier transform of $\\chi H$ and is bounded.\nIn the cone $\\abs{\\xi_1} \u003c \\abs{\\xi_2}$ around $(0, 1)$ one has $\\abs{\\xi} \u003c 2 \\abs{\\xi_2}$, and $\\hat\\chi(\\xi_2)$ decays faster than any power of $\\abs{\\xi_2}$.\nSo the point is microlocally regular, and it is not in $\\operatorname{WF}(u)$.\nIn fact $\\operatorname{WF}(u) = \\set{x_1 = 0} \\times \\set{(\\xi_1, 0) : \\xi_1 \\ne 0}$.\n\nThe wave front set transforms well under changes of coordinates, as a subset of the cotangent bundle minus its zero section, so the notion extends to distributions on manifolds.\n",
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