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  "body": "\n## Definition\n\nLet $n \\ge 1$ and $m \\in \\R$, and let $a \\in S^m(\\R^n \\times \\R^n)$ be a symbol of order $m$ in the [[symbol class|AM4RNWVV4Q]] $S^m = S^m_{1,0}$.\nFor $u$ in the [[Schwartz space|CG9JEATW3Q]] $\\mathcal{S}(\\R^n)$, let $\\hat{u}(\\xi) = \\int_{\\R^n} e^{-i x \\cdot \\xi} u(x) \\, dx$ be its [[Fourier transform|DPVM34ADZV]].\nThe *pseudo-differential operator* with symbol $a$ is the operator $\\mathrm{Op}(a) \\colon \\mathcal{S}(\\R^n) \\to \\mathcal{S}(\\R^n)$ given by\n$$\\mathrm{Op}(a) u(x) = \\frac{1}{(2\\pi)^n} \\int_{\\R^n} e^{i x \\cdot \\xi} a(x, \\xi) \\, \\hat{u}(\\xi) \\, d\\xi.$$\nAn operator of this form is a pseudo-differential operator *of order $m$*, and the set of them is denoted $\\Psi^m(\\R^n)$.\n\n## Remarks\n\nExample: let $P = \\sum_{\\abs{\\alpha} \\le k} a_\\alpha(x) D^\\alpha$ with $D^\\alpha = (-i)^{\\abs{\\alpha}} \\partial^\\alpha$ and coefficients $a_\\alpha$ smooth and bounded with all their derivatives.\nBy Fourier inversion, $D^\\alpha u(x) = (2\\pi)^{-n} \\int e^{i x \\cdot \\xi} \\xi^\\alpha \\hat{u}(\\xi) \\, d\\xi$, so $P = \\mathrm{Op}(p)$ with $p(x, \\xi) = \\sum_\\alpha a_\\alpha(x) \\xi^\\alpha \\in S^k$.\n\nNon-example: the operator $u \\mapsto \\mathcal{F}^{-1}\\big( e^{i\\abs{\\xi}^2} \\hat{u} \\big)$, a Schrödinger evolution operator, is not a pseudo-differential operator of any order.\nThe symbol of an operator of this form is determined by the operator, here $e^{i \\abs{\\xi}^2}$, and it is in no class $S^m$.\n\nThe composition of $\\mathrm{Op}(a)$ with $a \\in S^m$ and $\\mathrm{Op}(b)$ with $b \\in S^{m'}$ is $\\mathrm{Op}(c)$ with $c \\in S^{m + m'}$, and $c - ab \\in S^{m + m' - 1}$.\nOperators of order $0$ are bounded on $L^2(\\R^n)$, and operators of order $-\\infty$ are smoothing.\nThe same formula with a symbol in $S^m_{\\rho, \\delta}$ defines the larger classes $\\Psi^m_{\\rho, \\delta}$.\n",
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