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  "body": "\n## Summary\n\nThe wave front set of a distribution records where the distribution fails to be smooth, and in which frequency directions it fails.\nIt is a closed subset of $\\Omega \\times (\\R^n \\setminus \\set{0})$ that is conic in the second variable.\nIts projection to $\\Omega$ is the singular support, the set of points near which the distribution is not a smooth function.\nIt is the basic object of [[microlocal analysis|5B6WEQX7XK]] and the language of [[propagation of singularities|P2QJ7NPW3Y]].\nSee the [[definition|PF3G0J5JD6]].\n\n## Explanations\n\n### L1 human\n\nA function with a jump along a line is rough at the points of the line, but only in one direction: across the line.\nAlong the line it looks perfectly regular.\nThe wave front set keeps track of this: at each bad point it lists the directions in which the function oscillates without control.\nOne finds these directions by cutting the function off near the point and checking in which directions the [[Fourier transform|DPVM34ADZV]] of the piece decays slowly.\nFor the jump function $H(x_1)$ in the plane, the bad points form the line $x_1 = 0$, and the bad directions there are $(\\pm 1, 0)$, the two normal directions.\n",
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