Bessel potential space

Summary

A Bessel potential space $H^{s,p}(\R^n)$ consists of the functions or tempered distributions whose "derivative of order $s$" lies in $L^p$. The 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. These spaces extend the Sobolev spaces $W^{k,p}$ to fractional and negative orders, and for $p = 2$ they are the fractional Sobolev spaces $H^s$. They are a standard scale of spaces for elliptic equations and for interpolation.