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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $m \\in \\R$, and let $0 \\le \\delta \\le \\rho \\le 1$ with $\\delta \u003c 1$.\nThe *symbol class* $S^m_{\\rho, \\delta}(\\R^n \\times \\R^n)$ is the set of [[smooth functions|0V33BNKZ9S]] $a \\colon \\R^n \\times \\R^n \\to \\C$, $(x, \\xi) \\mapsto a(x, \\xi)$, such that for all multi-indices $\\alpha, \\beta \\in \\N^n$ there is a constant $C_{\\alpha\\beta}$ with\n$$\\abs{\\partial_\\xi^\\alpha \\partial_x^\\beta a(x, \\xi)} \\le C_{\\alpha\\beta} \\, (1 + \\abs{\\xi})^{m - \\rho \\abs{\\alpha} + \\delta \\abs{\\beta}} \\quad \\text{for all } x, \\xi \\in \\R^n.$$\nHere $\\partial_\\xi^\\alpha$ and $\\partial_x^\\beta$ are iterated [[partial derivatives|09EGC6F8PT]] and $\\abs{\\alpha} = \\alpha_1 + \\cdots + \\alpha_n$.\nThe elements of $S^m_{\\rho, \\delta}$ are *symbols of order $m$* and *type $(\\rho, \\delta)$*.\nOne writes $S^m = S^m_{1, 0}$ and $S^{-\\infty} = \\bigcap_{m \\in \\R} S^m$.\n\n## Remarks\n\nExample: let $a(x, \\xi) = \\sum_{\\abs{\\gamma} \\le k} a_\\gamma(x) \\xi^\\gamma$, where each $a_\\gamma$ is smooth and bounded with all its derivatives.\nThen $a \\in S^k$.\nIndeed $\\partial_\\xi^\\alpha \\xi^\\gamma$ is zero or a monomial of degree $\\abs{\\gamma} - \\abs{\\alpha}$, so it is bounded by a constant times $(1 + \\abs{\\xi})^{k - \\abs{\\alpha}}$.\nDerivatives in $x$ only change the bounded coefficients.\nAlso $(1 + \\abs{\\xi}^2)^{s/2} \\in S^s$ for every $s \\in \\R$.\n\nNon-example: $a(x, \\xi) = \\abs{\\xi}$ is not in any symbol class, because it is not smooth at $\\xi = 0$.\nSimilarly $a(\\xi) = e^{i \\abs{\\xi}^2}$ is smooth but lies in no class $S^m_{1,0}$, because its derivatives of order $k$ in $\\xi$ grow like $\\abs{\\xi}^k$.\n\nThe classes are nested: $S^m_{\\rho, \\delta} \\subseteq S^{m'}_{\\rho, \\delta}$ for $m \\le m'$.\nProducts satisfy $S^m_{\\rho, \\delta} \\cdot S^{m'}_{\\rho, \\delta} \\subseteq S^{m + m'}_{\\rho, \\delta}$, and $\\partial_\\xi^\\alpha$ maps $S^m_{\\rho, \\delta}$ into $S^{m - \\rho\\abs{\\alpha}}_{\\rho, \\delta}$.\n",
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