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  "body": "\n## Summary\n\nThe Bessel potential of order $\\alpha \u003e 0$ is the operator $(I - \\Delta)^{-\\alpha/2}$ on functions on $\\R^n$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]].\nIt is a convolution with a positive integrable kernel $G_\\alpha$.\nNear the origin $G_\\alpha$ behaves like the kernel of the [[Riesz potential|1V2N8MFSHC]].\nIt decays exponentially at infinity, which removes the problems of the Riesz potential at large scales.\nThe images of $L^p$ under Bessel potentials are the [[Bessel potential spaces|DVJM7NBSF2]], which give Sobolev spaces of fractional order.\n\n## Explanations\n\n### L1 human\n\nThe Bessel potential smooths a function by a fractional amount $\\alpha$, much like the Riesz potential.\nThe difference is the extra $I$ in $(I - \\Delta)$, which acts like a mass term in physics and makes the smoothing kernel die off quickly far away.\nIn one dimension with $\\alpha = 2$ the kernel is $\\frac12 e^{-\\abs{x}}$, so the Bessel potential averages $f$ with an exponentially decaying weight.\nThis operator solves $u - u'' = f$.\n",
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