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  "body": "\n## Summary\n\nA Bessel potential space $H^{s,p}(\\R^n)$ consists of the functions or tempered distributions whose \"derivative of order $s$\" lies in $L^p$.\nThe derivative of order $s$ is defined on the Fourier side by multiplication with $(1 + \\abs{\\xi}^2)^{s/2}$, so $s$ may be any real number.\nThese spaces extend the [[Sobolev spaces|M4CEZ5K9HP]] $W^{k,p}$ to fractional and negative orders, and for $p = 2$ they are the [[fractional Sobolev spaces|N6NX1VT837]] $H^s$.\nThey are a standard scale of spaces for elliptic equations and for interpolation.\n\n## Explanations\n\n### L1 human\n\nDifferentiating a function multiplies its Fourier transform by the frequency.\nSo a function is smooth when its Fourier transform still has small size after multiplication by a power of the frequency.\nThe Bessel potential space of order $s$ asks exactly this, with the power $(1 + \\abs{\\xi}^2)^{s/2}$ and with size measured in $L^p$.\nFor $s = 0$ the space is just $L^p$.\nFor $s = 1$ and $1 \u003c p \u003c \\infty$ it is the space of $L^p$ functions with first derivatives in $L^p$.\n",
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