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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $a \\colon \\R^n \\times \\R^n \\to \\C$ be a smooth function such that every derivative of $a$ is bounded: for all multi-indices $\\alpha, \\beta$,\n$$\\sup_{x, \\xi \\in \\R^n} \\abs{\\partial_x^\\alpha \\partial_\\xi^\\beta a(x, \\xi)} \u003c \\infty.$$\nThat is, $a$ lies in the [[symbol class|AM4RNWVV4Q]] $S^0_{0,0}$.\nLet $a(x, D)$ be the [[pseudo-differential operator|FF27N0BWTP]] with symbol $a$, defined on Schwartz functions $u$ by\n$$a(x, D) u(x) = (2\\pi)^{-n} \\int_{\\R^n} e^{i x \\cdot \\xi} a(x, \\xi) \\, \\hat u(\\xi) \\, d\\xi,$$\nwhere $\\hat u$ is the [[Fourier transform|DPVM34ADZV]] of $u$.\nThen $a(x, D)$ extends to a bounded operator on $L^2(\\R^n)$.\nMoreover there are a constant $C$ and an integer $N$, depending only on $n$, with\n$$\\norm{a(x, D)}_{L^2 \\to L^2} \\le C \\max_{\\abs{\\alpha} \\le N,\\ \\abs{\\beta} \\le N} \\ \\sup_{x, \\xi} \\abs{\\partial_x^\\alpha \\partial_\\xi^\\beta a(x, \\xi)}.$$\n\n## Remarks\n\nOne standard proof cuts the symbol into pieces $a_{jk}$ localized near the points $(j, k)$ of the lattice $\\Z^n \\times \\Z^n$ by a smooth partition of unity.\nEach piece gives an operator with norm bounded by finitely many derivatives of $a$.\nIntegration by parts shows that the products $T_{jk}^* T_{j'k'}$ and $T_{jk} T_{j'k'}^*$ decay fast in $\\abs{j - j'}$ and $\\abs{k - k'}$.\nThe [[Cotlar–Stein lemma|XBJEDC18B1]] then sums the pieces.\n\nExample: if $a(x, \\xi) = m(\\xi)$ does not depend on $x$, then $a(x, D)$ is the Fourier multiplier with symbol $m$, and Plancherel's theorem gives the bound $\\sup \\abs{m}$ directly.\nThe theorem says that bounded smooth dependence on $x$ costs only finitely many derivatives.\nThe same conclusion holds for symbols in the classes $S^0_{\\rho, \\rho}$ with $0 \\le \\rho \u003c 1$.\n\nFor the class $S^0_{1,1}$, which allows $x$-derivatives to grow like $\\abs{\\xi}$, the operators are in general not bounded on $L^2$.\n\n## Explanations\n\n### L1 human\n\nA pseudo-differential operator acts on a function by changing the strength of each frequency $\\xi$, by an amount $a(x, \\xi)$ that may also depend on the position $x$.\nWhen $a$ depends only on $\\xi$, the operator is bounded on $L^2$ as long as $a$ is bounded.\nThe Calderón–Vaillancourt theorem says the operator stays bounded on $L^2$ when $a$ depends on both variables, provided $a$ and its derivatives are all bounded.\n\n### L2 human\n\nThis is the basic $L^2$ boundedness result of the calculus of pseudo-differential operators.\nIt shows that operators of order $0$ in the classes $S^0_{\\rho,\\delta}$ with $\\delta \\le \\rho$ and $\\delta \u003c 1$ are bounded on $L^2$.\nBy composition with $(I - \\Delta)^{s/2}$, operators of order $m$ then map the Sobolev space $H^{s+m}$ to $H^s$.\nIts proof is a model use of almost orthogonality.\nThe class $S^0_{0,0}$ is invariant under translations in both $x$ and $\\xi$, which makes it natural in phase-space and semiclassical analysis.\n",
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