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  "body": "\n## Summary\n\nFractional integration on $\\R^n$ is the operator $I_\\alpha$, for $0 \u003c \\alpha \u003c n$, that convolves a function with the kernel $\\abs{x}^{\\alpha - n}$, suitably normalized.\nIt is also called the Riesz potential.\nOn the [[Fourier transform|DPVM34ADZV]] side it multiplies by a constant times $\\abs{\\xi}^{-\\alpha}$, so it is the inverse of the [[fractional Laplacian|F8RSNQBMBE]] $(-\\Delta)^{\\alpha/2}$.\nIt is the Euclidean analogue of the [[Riemann–Liouville integral|JTKCZ44NN3]], and it is the main tool behind the Hardy–Littlewood–Sobolev and Sobolev inequalities.\n\n## Explanations\n\n### L1 human\n\nIntegrating a function once makes it smoother by one order; fractional integration makes it smoother by a fractional order $\\alpha$.\nIn $\\R^n$ this is done by averaging $f$ against the weight $\\abs{x - y}^{\\alpha - n}$, which is large near $x$ and decays far away.\nFor $n = 3$ and $\\alpha = 2$ the result is the Newtonian potential $\\frac{1}{4\\pi}\\int \\frac{f(y)}{\\abs{x - y}} \\, dy$, the gravitational potential of a mass density $f$.\nDoing it twice, of orders $\\alpha$ and $\\beta$, is the same as doing it once of order $\\alpha + \\beta$, as long as $\\alpha + \\beta \u003c n$.\n",
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