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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\alpha$ be a real number with $0 \u003c \\alpha \u003c n$.\nLet\n$$c_\\alpha = \\frac{\\pi^{n/2} \\, 2^\\alpha \\, \\Gamma(\\alpha/2)}{\\Gamma((n - \\alpha)/2)},$$\nwhere $\\Gamma$ is the [[gamma function|0H5SYBSED9]].\nLet $f$ be a measurable function on $\\R^n$.\nAt each point $x \\in \\R^n$ where the integral converges absolutely, the *Riesz potential* of order $\\alpha$ of $f$ is\n$$I_\\alpha f(x) = \\frac{1}{c_\\alpha} \\int_{\\R^n} \\frac{f(y)}{\\abs{x - y}^{n - \\alpha}} \\, dy.$$\nThe integral is taken with respect to [[Lebesgue measure|2Z2QVWZ262]].\nThe map $f \\mapsto I_\\alpha f$ is the *Riesz potential operator* $I_\\alpha$.\n\n## Remarks\n\nIf $f \\in L^p(\\R^n)$ with $1 \\le p \u003c n/\\alpha$, then $I_\\alpha f(x)$ converges absolutely for almost every $x$.\nFor $1 \u003c p \u003c n/\\alpha$ and $1/q = 1/p - \\alpha/n$, the [[Hardy–Littlewood–Sobolev inequality|HG65C9R1SP]] gives $\\norm{I_\\alpha f}_q \\le A \\norm{f}_p$ with $A$ depending only on $n$, $\\alpha$ and $p$, see [[Lp space|SASP14MA1Z]].\n\nWith the Fourier transform $\\hat f(\\xi) = \\int_{\\R^n} f(x) e^{-2\\pi i x \\cdot \\xi} \\, dx$, the constant $c_\\alpha$ is chosen so that $\\widehat{I_\\alpha f}(\\xi) = (2\\pi \\abs{\\xi})^{-\\alpha} \\hat f(\\xi)$ for nice $f$.\nSo $I_\\alpha = (-\\Delta)^{-\\alpha/2}$, the inverse of the [[fractional Laplacian|F8RSNQBMBE]], and $I_\\alpha I_\\beta = I_{\\alpha + \\beta}$ when $\\alpha + \\beta \u003c n$.\n\nExample: $n = 3$ and $\\alpha = 2$.\nThen $c_2 = \\pi^{3/2} \\cdot 4 \\cdot \\Gamma(1) / \\Gamma(1/2) = 4\\pi$, since $\\Gamma(1/2) = \\sqrt{\\pi}$.\nSo $I_2 f(x) = \\frac{1}{4\\pi} \\int_{\\R^3} \\frac{f(y)}{\\abs{x - y}} \\, dy$, the Newtonian potential, which solves $-\\Delta u = f$ for smooth $f$ with compact support.\n\nNon-example: for $f = 1$ the integral $\\int_{\\R^n} \\abs{x - y}^{\\alpha - n} \\, dy$ diverges at every $x$, because $\\alpha - n \u003e -n$, so $I_\\alpha 1$ is not defined.\n\n## Explanations\n\n### L1 human\n\nThe Riesz potential averages $f$ with a weight that is large near the point $x$ and decays like a power of the distance.\n",
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