---
tag: N6NX1VT837
kind: concept
name: Fractional Sobolev space
msc: ["46E35", 35R11]
sources:
  - url: https://en.wikipedia.org/wiki/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|DPVM34ADZV]].
These spaces fill the gaps between the integer [[Sobolev spaces|M4CEZ5K9HP]].
They describe the boundary values of Sobolev functions and are the natural energy spaces for nonlocal operators such as the fractional Laplacian.

## Explanations

### L1 human

An ordinary derivative measures how fast a function changes at a point.
A 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.
So $W^{s,p}$ is an averaged version of the Hölder condition with exponent $s$.
For example, on an interval, a function with a jump lies in $W^{s,2}$ exactly when $s < 1/2$.