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  "body": "\n## Definition\n\nLet $X$ be an open subset of $\\R^n$, or a smooth manifold, with cotangent bundle $T^*X$.\n*Microlocal analysis* studies distributions and linear operators on $X$ through properties localized at points $(x, \\xi) \\in T^*X \\setminus 0$.\nHere $x$ is a position and $\\xi \\neq 0$ is a frequency direction.\nA property holds *microlocally* near $(x_0, \\xi_0)$ if it holds after cutting off near $x_0$ and keeping only frequencies in a conic neighbourhood of $\\xi_0$.\nThe basic notions are:\n\n1. The [[wave front set|3T5Q9WVQX8]] $\\mathrm{WF}(u) \\subset T^*X \\setminus 0$ of a distribution $u$: the set of $(x_0, \\xi_0)$ near which $u$ fails to be smooth microlocally.\n2. Properly supported [[pseudo-differential operators|FF27N0BWTP]], which act microlocally and satisfy $\\mathrm{WF}(Pu) \\subseteq \\mathrm{WF}(u)$.\n\n## Remarks\n\nExample: let $u = H(x_1)$ on $\\R^2$, where $H$ is the Heaviside function.\nThen $u$ is smooth off the line $x_1 = 0$, and $\\mathrm{WF}(u) = \\set{(x, \\xi) : x_1 = 0, \\ \\xi = (\\xi_1, 0), \\ \\xi_1 \\neq 0}$.\nThe check: take a cutoff $\\chi(x)$ near a point of the line.\nThe [[Fourier transform|DPVM34ADZV]] of $\\chi u$ decays rapidly in every closed cone avoiding the $\\xi_1$-axis, but only like $\\abs{\\xi_1}^{-1}$ along it.\nNon-example: classical local regularity, such as \"$u$ is smooth near $x_0$\", is not a microlocal statement, since it does not distinguish frequency directions.\nThe projection of $\\mathrm{WF}(u)$ to $X$ is the singular support: $\\pi(\\mathrm{WF}(u)) = \\operatorname{sing\\,supp} u$.\n",
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