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  "body": "\n## Summary\n\nThe Riesz potential of order $\\alpha$, for $0 \u003c \\alpha \u003c n$, is the operator $I_\\alpha$ that convolves a function on $\\R^n$ with a constant multiple of $\\abs{x}^{\\alpha - n}$.\nIt is the inverse of the [[fractional Laplacian|F8RSNQBMBE]] $(-\\Delta)^{\\alpha/2}$, so it integrates a function to the fractional order $\\alpha$.\nFor $\\alpha = 2$ and $n \\ge 3$ it is, up to a constant factor, the [[Newtonian potential|DCMFQ01QGS]], which solves Poisson's equation.\nIts main estimate is the [[Hardy–Littlewood–Sobolev inequality|HG65C9R1SP]], which underlies the Sobolev embedding theorem.\n\n## Explanations\n\n### L1 human\n\nTaking a derivative makes a function rougher, and integrating makes it smoother.\nThe Riesz potential is a way to integrate \"a fraction of a time\" in several variables.\nIt averages the function with a weight $\\abs{x - y}^{\\alpha - n}$ that is large near $x$ and decays with distance.\nThe larger $\\alpha$ is, the slower the decay and the smoother the result.\nIn $\\R^3$ with $\\alpha = 2$ the weight is $1/\\abs{x - y}$, the gravitational potential of a point mass.\n",
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