Paper
Higher-order affine Sobolev inequalities
Authors
Tristan Bullion-Gauthier
Abstract
Zhang refined the classical Sobolev inequality , where , by replacing with a smaller quantity invariant by unimodular affine transformations. The analogue result in homogeneous fractional Sobolev spaces , with and , was obtained by Haddad and Ludwig. We generalize their results to the case where . Our approach, based on the existence of optimal unimodular transformations, allows us to obtain various affine inequalities, such as affine Sobolev inequalities, reverse affine inequalities, and affine Gagliardo-Nirenberg type inequalities. In a different but related direction, we also answer a question concerning reverse affine inequalities, raised by Haddad, Jiménez, and Montenegro.