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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $0 \u003c \\alpha \u003c n$.\nLet $\\Gamma$ be the [[gamma function|0H5SYBSED9]] and let\n$$\\gamma(\\alpha) = \\frac{\\pi^{n/2} \\, 2^\\alpha \\, \\Gamma(\\alpha/2)}{\\Gamma((n - \\alpha)/2)}.$$\nLet $f$ be a measurable function on $\\R^n$.\nIts *fractional integral*, or *Riesz potential*, of order $\\alpha$ is\n$$I_\\alpha f(x) = \\frac{1}{\\gamma(\\alpha)} \\int_{\\R^n} \\frac{f(y)}{\\abs{x - y}^{n - \\alpha}} \\, dy,$$\nat each point $x \\in \\R^n$ where the [[Lebesgue integral|DWJF7ABPR1]] converges absolutely.\nThe integral is with respect to [[Lebesgue measure|2Z2QVWZ262]].\n\n## Remarks\n\nIf $f \\in L^p(\\R^n)$, an element of the [[Lp space|SASP14MA1Z]], 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 $\\gamma(\\alpha)$ is chosen so that $\\widehat{I_\\alpha f}(\\xi) = (2\\pi\\abs{\\xi})^{-\\alpha} \\hat f(\\xi)$, for example for Schwartz functions $f$.\nIn this sense $I_\\alpha = (-\\Delta)^{-\\alpha/2}$, and $I_\\alpha I_\\beta = I_{\\alpha + \\beta}$ when $\\alpha + \\beta \u003c n$.\n\nExample: $n = 3$ and $\\alpha = 2$.\nThen $\\gamma(2) = \\pi^{3/2} \\cdot 4 \\cdot \\Gamma(1)/\\Gamma(1/2) = 4\\pi$, since $\\Gamma(1/2) = \\sqrt{\\pi}$.\nSo $I_2 f(x) = \\frac{1}{4\\pi}\\int_{\\R^3} \\frac{f(y)}{\\abs{x - y}} \\, dy$ is the Newtonian potential, and $-\\Delta I_2 f = f$ for smooth $f$ with compact support.\n\nNon-example: the order $\\alpha = n$ fails the condition $0 \u003c \\alpha \u003c n$.\nThen the kernel $\\abs{x - y}^{0}$ is the constant $1$, so the formula gives $\\int_{\\R^n} f(y) \\, dy$, which does not depend on $x$ and smooths nothing.\nAlso $\\Gamma((n - \\alpha)/2) = \\Gamma(0)$ is undefined, so the normalization breaks down.\n",
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