Affine fractional Sobolev and isoperimetric inequalities

Abstract

Sharp affine fractional Sobolev inequalities for functions on Rⁿ are established. For each 0 < s < 1, the new inequalities are significantly stronger than (and directly imply) the sharp fractional Sobolev inequalities of Almgren and Lieb. In the limit as s → 1⁻, the new inequalities imply the sharp affine Sobolev inequality of Gaoyong Zhang. As a consequence, fractional Petty projection inequalities are obtained which are stronger than the fractional Euclidean isoperimetric inequalities, and a natural conjecture for radial mean bodies is proved.