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  "body": "\n## Summary\n\nA fractional Sobolev space $W^{s,p}$ measures smoothness of a non-integer order $s$ in the $L^p$ sense.\nFor $0 \u003c s \u003c 1$ it is defined by the Gagliardo seminorm, a double integral of difference quotients.\nFor $p = 2$ on $\\R^n$ it agrees with the space $H^s$ defined by the weight $(1 + \\abs{\\xi}^2)^{s}$ on the [[Fourier transform|DPVM34ADZV]].\nThese spaces fill the gaps between the integer [[Sobolev spaces|M4CEZ5K9HP]].\nThey describe the boundary values of Sobolev functions and are the natural energy spaces for nonlocal operators such as the fractional Laplacian.\n\n## Explanations\n\n### L1 human\n\nAn ordinary derivative measures how fast a function changes at a point.\nA fractional derivative of order $s$ between $0$ and $1$ asks for something weaker: the difference $u(x) - u(y)$ must be small compared with $\\abs{x - y}^s$, on average over all pairs of points.\nSo $W^{s,p}$ is an averaged version of the Hölder condition with exponent $s$.\nFor example, on an interval, a function with a jump lies in $W^{s,2}$ exactly when $s \u003c 1/2$.\n",
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