Fractional Sobolev space

Summary

A fractional Sobolev space $W^{s,p}$ measures smoothness of a non-integer order $s$ in the $L^p$ sense. For $0 < s < 1$ it is defined by the Gagliardo seminorm, a double integral of difference quotients. For $p = 2$ on $\R^n$ it agrees with the space $H^s$ defined by the weight $(1 + \abs{\xi}^2)^{s}$ on the Fourier transform. These spaces fill the gaps between the integer Sobolev spaces. They describe the boundary values of Sobolev functions and are the natural energy spaces for nonlocal operators such as the fractional Laplacian.