  Sharp affine fractional $L^p$ Sobolev inequalities for functions on $\mathbb
R^n$ are established. The new inequalities are stronger than (and directly
imply) the sharp fractional $L^p$ Sobolev inequalities. They are fractional
versions of the affine $L^p$ Sobolev inequalities of Lutwak, Yang, and Zhang.
In addition, affine fractional asymmetric $L^p$ Sobolev inequalities are
established.
