# Affine vs. Euclidean Sobolev inequalities

## Franz Schuster, TU Vienna

## October 31, 2017

In this talk we explain how every even, zonal measure on the Euclidean
unit sphere gives rise to a sharp Sobolev inequality for functions of
bounded variation which directly implies the classical Euclidean
Sobolev inequality. The strongest member of this large family of
inequalities is shown to be the only affine invariant one among them —
the affine Zhang–Sobolev inequality. We also relate our new Sobolev
inequalities to the sharp Gromov–Gagliardo–Nirenberg Sobolev
inequality for general norms and discuss further improvements of
special cases.