Affine fractional $L^p$ Sobolev inequalities

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.