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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\mathcal{F} \\colon L^2(\\R^n) \\to L^2(\\R^n)$ be the [[Fourier transform|DPVM34ADZV]] on $L^2$, extended from $\\mathcal{F} f(\\xi) = \\int_{\\R^n} f(x) e^{-2\\pi i x \\cdot \\xi}\\, dx$ by the [[Plancherel theorem|HPZHRZ0502]].\nLet $m \\colon \\R^n \\to \\C$ be a bounded measurable function.\nThe *Fourier multiplier* with symbol $m$ is the operator $T_m \\colon L^2(\\R^n) \\to L^2(\\R^n)$ given by\n$$T_m f = \\mathcal{F}^{-1}\\big( m \\cdot \\mathcal{F} f \\big).$$\n\n## Remarks\n\nBy Plancherel, $T_m$ is a bounded linear operator on [[$L^2(\\R^n)$|SASP14MA1Z]] with operator norm equal to $\\norm{m}_{L^\\infty}$.\nIt commutes with every translation, and conversely every bounded operator on $L^2(\\R^n)$ that commutes with all translations is $T_m$ for a unique $m \\in L^\\infty(\\R^n)$.\n\nExample: on $\\R$, the symbol $m(\\xi) = -i \\operatorname{sgn}(\\xi)$ is bounded, and $T_m$ is the [[Hilbert transform|NSCEBXM7ZM]].\nExample: for $t \u003e 0$, the symbol $m(\\xi) = e^{-4\\pi^2 t \\abs{\\xi}^2}$ gives the heat semigroup: $u(t, \\cdot) = T_m f$ solves $\\partial_t u = \\Delta u$ with $u(0, \\cdot) = f$, because $\\Delta$ acts on the Fourier side as multiplication by $-4\\pi^2 \\abs{\\xi}^2$.\n\nNon-example: the symbol $m(\\xi) = 2\\pi i \\xi$ on $\\R$ is not bounded; the corresponding operator is differentiation, which is not defined on all of $L^2(\\R)$.\nIt is still a Fourier multiplier in the wider sense of an operator on the Schwartz space or on tempered distributions.\n\nFor the torus $\\mathbb{T}^n$ the same definition uses a bounded sequence $(m(k))_{k \\in \\Z^n}$ that multiplies the Fourier coefficients.\nThe symbol $m$ is called an $L^p$ multiplier when $T_m$, restricted to $L^2 \\cap L^p$, extends to a bounded operator on $L^p$.\n",
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