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  "body": "\n## Summary\n\nA pseudo-differential operator is an operator of the form $u \\mapsto \\int e^{i x \\cdot \\xi} a(x, \\xi) \\hat{u}(\\xi) \\, d\\xi / (2\\pi)^n$, where the symbol $a$ belongs to a [[symbol class|AM4RNWVV4Q]].\nLinear differential operators with smooth bounded coefficients are the case where $a$ is a polynomial in $\\xi$.\nPseudo-differential operators form an algebra that also contains approximate inverses of elliptic operators.\nThey are a basic tool of microlocal analysis and of the theory of linear partial differential equations.\n\n## Explanations\n\n### L1 human\n\nThe [[Fourier transform|DPVM34ADZV]] turns differentiation into multiplication: $\\partial_x$ becomes multiplication by $i\\xi$.\nSo a differential operator is the same as multiplication by a polynomial in frequency space.\nA pseudo-differential operator allows other functions of frequency, such as $\\sqrt{1 + \\xi^2}$, and lets them depend on position too.\nThis makes it possible to divide by a symbol, and so to invert an elliptic operator up to a small error.\nFor example, $(1 - \\Delta)^{-1}$ is the pseudo-differential operator with symbol $(1 + \\abs{\\xi}^2)^{-1}$.\n",
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