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  "body": "\n## Summary\n\nThe Riesz potential operator $I_\\alpha$ of order $\\alpha$, with $0 \u003c \\alpha \u003c n$, is the integral operator on $\\R^n$ with kernel a constant times $\\abs{x - y}^{\\alpha - n}$.\nIt is the inverse of the fractional power $(-\\Delta)^{\\alpha/2}$ of the [[Laplacian|0E0B6RA8K0]], so it acts as integration of order $\\alpha$ in $n$ variables.\nFor $n = 3$ and $\\alpha = 2$ it is the Newtonian potential of physics.\nIts mapping properties between [[Lp spaces|SASP14MA1Z]], given by the [[Hardy–Littlewood–Sobolev inequality|HG65C9R1SP]], underlie the Sobolev embedding theorems.\n\n## Explanations\n\n### L1 human\n\nThe gravitational potential of a mass distribution in space is obtained by adding up contributions $1/\\abs{x - y}$ from each piece of mass at $y$.\nThe Riesz potential replaces the power $1/\\abs{x - y}$ by $\\abs{x - y}^{\\alpha - n}$ in $n$ dimensions.\nSmall $\\alpha$ gives a sharply peaked kernel that smooths a function only a little, and larger $\\alpha$ smooths more.\nIn one dimension with $0 \u003c \\alpha \u003c 1$, it is a two-sided version of the [[Riemann–Liouville integral|JTKCZ44NN3]] of order $\\alpha$.\n",
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