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  "body": "\n## Summary\n\nMicrolocal analysis studies functions, distributions and linear operators on $\\R^n$ or on a manifold by localizing both in position $x$ and in frequency direction $\\xi$.\nIts natural home is phase space, the cotangent bundle, whose points are pairs $(x, \\xi)$.\nIts main tools are the [[wave front set|3T5Q9WVQX8]], [[pseudo-differential operators|FF27N0BWTP]] and [[Fourier integral operators|EVS4JPEE45]].\nIt explains where and in which directions solutions of linear [[partial differential equations|GC5BDZ6QBC]] are singular, and how these singularities move.\n\n## Explanations\n\n### L1 human\n\nA function can be smooth in some places and rough in others.\nMicrolocal analysis also records the direction in which it is rough.\nPicture a function on the plane that jumps across the line $x_1 = 0$: it is singular only on that line, and only in the directions $\\pm(1, 0)$ normal to it.\nThe [[Fourier transform|DPVM34ADZV]] of the function, cut off near a point of the line, decays fast in every direction except those two.\nThis finer picture shows, for example, that singularities of waves travel along rays, as in [[propagation of singularities|P2QJ7NPW3Y]].\n",
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