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  "body": "\n## Definition\n\nLet $n \\ge 1$ and $0 \u003c s \u003c 1$.\nLet $\\mathcal{S}(\\R^n)$ be the [[Schwartz space|CG9JEATW3Q]], and let $\\hat u(\\xi) = \\int_{\\R^n} e^{-i x \\cdot \\xi} u(x) \\, dx$ be the [[Fourier transform|DPVM34ADZV]] of $u$.\nThe *fractional Laplacian* of $u \\in \\mathcal{S}(\\R^n)$ is the function $(-\\Delta)^s u$ defined by\n$$\\widehat{(-\\Delta)^s u}(\\xi) = \\abs{\\xi}^{2s} \\hat u(\\xi), \\qquad \\xi \\in \\R^n.$$\n\n## Remarks\n\nEquivalently, for $u \\in \\mathcal{S}(\\R^n)$ and $x \\in \\R^n$,\n$$(-\\Delta)^s u(x) = C_{n,s} \\lim_{\\varepsilon \\to 0^+} \\int_{\\abs{x - y} \u003e \\varepsilon} \\frac{u(x) - u(y)}{\\abs{x - y}^{n + 2s}} \\, dy, \\qquad C_{n,s} = \\frac{4^s \\, \\Gamma(n/2 + s)}{\\pi^{n/2} \\abs{\\Gamma(-s)}}.$$\nThe integral formula also makes sense for bounded $C^2$ functions.\nExample: for $u(x) = \\cos(k \\cdot x)$ with $k \\in \\R^n$, the integral formula gives $(-\\Delta)^s u = \\abs{k}^{2s} u$, in agreement with the multiplier $\\abs{\\xi}^{2s}$.\nNon-example: the exponent $s = 1$ is excluded, since the constant $C_{n,s}$ tends to $0$ as $s \\to 1$ and the singular integral diverges; the multiplier $\\abs{\\xi}^2$ gives the ordinary operator $-\\Delta$ instead.\nThe function $(-\\Delta)^s u$ is smooth but in general not in $\\mathcal{S}(\\R^n)$, since $\\abs{\\xi}^{2s}$ is not smooth at $\\xi = 0$; it decays like $\\abs{x}^{-n-2s}$.\n",
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