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  "body": "\n## Definition\n\nLet $n \\ge 1$ and $0 \u003c \\alpha \u003c n$.\nLet $\\Gamma$ be the [[gamma function|0H5SYBSED9]] and let\n$$c_\\alpha = \\pi^{n/2} \\, 2^\\alpha \\, \\frac{\\Gamma(\\alpha/2)}{\\Gamma((n - \\alpha)/2)}.$$\nFor a measurable function $f$ on $\\R^n$, the *Riesz potential* of order $\\alpha$ of $f$ is\n$$I_\\alpha f(x) = \\frac{1}{c_\\alpha} \\int_{\\R^n} \\frac{f(y)}{\\abs{x - y}^{n - \\alpha}} \\, dy,$$\nat each $x \\in \\R^n$ where the integral converges absolutely.\n\n## Remarks\n\nSo $I_\\alpha f$ is the [[convolution|2Z29JN3VES]] of $f$ with the kernel $\\abs{x}^{\\alpha - n} / c_\\alpha$.\nIf $f \\in L^p(\\R^n)$ with $1 \\le p \u003c n/\\alpha$, the integral converges absolutely for almost every $x$.\nWith the [[Fourier transform|DPVM34ADZV]] $\\hat f(\\xi) = \\int f(x) e^{-2\\pi i x \\cdot \\xi} \\, dx$, the constant $c_\\alpha$ is chosen so that $\\widehat{I_\\alpha f}(\\xi) = (2\\pi \\abs{\\xi})^{-\\alpha} \\hat f(\\xi)$ for Schwartz functions $f$, that is, $I_\\alpha = (-\\Delta)^{-\\alpha/2}$.\nThe semigroup law $I_\\alpha I_\\beta = I_{\\alpha + \\beta}$ holds for $\\alpha + \\beta \u003c n$.\n\nExample: for $n \\ge 3$ and $\\alpha = 2$, one has $c_2 = 4 \\pi^{n/2} / \\Gamma(n/2 - 1) = (n - 2) \\sigma_{n-1}$, where $\\sigma_{n-1} = 2 \\pi^{n/2} / \\Gamma(n/2)$ is the area of the unit sphere.\nIndeed $\\Gamma(n/2) = (n/2 - 1) \\Gamma(n/2 - 1)$, so $(n - 2) \\cdot 2 \\pi^{n/2} / \\Gamma(n/2) = 4 \\pi^{n/2} / \\Gamma(n/2 - 1)$.\nSo $I_2 f(x) = \\frac{1}{(n-2) \\sigma_{n-1}} \\int f(y) \\abs{x - y}^{2-n} \\, dy$, and $-\\Delta I_2 f = f$ for smooth $f$ with compact support.\n\nNon-example: the constant function $f = 1$ has no Riesz potential.\nFor every $x$ the integral $\\int \\abs{x - y}^{\\alpha - n} \\, dy$ diverges, since $\\abs{y}^{\\alpha - n}$ is not integrable at infinity when $\\alpha \u003e 0$.\n",
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  "name": "Riesz potential",
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