Higher-order affine Sobolev inequalities

TL;DR

This paper extends affine Sobolev inequalities to higher-order fractional spaces, introduces new affine inequalities, and addresses open questions on reverse affine inequalities, broadening the theoretical understanding of affine invariance in Sobolev spaces.

Contribution

It generalizes affine Sobolev inequalities to the case where s > 1, introduces various new affine inequalities, and resolves an open question on reverse affine inequalities.

Findings

Extended affine Sobolev inequalities to s > 1

Derived new affine Gagliardo-Nirenberg inequalities

Resolved an open question on reverse affine inequalities

Abstract

Zhang refined the classical Sobolev inequality $\|f\|_{L^{Np/(N-p)}} \lesssim \| \nabla f \|_{L^p}$, where $1\leq p \lt N$, by replacing $\|\nabla f\|_{L^p}$ with a smaller quantity invariant by unimodular affine transformations. The analogue result in homogeneous fractional Sobolev spaces $\mathring{W}^{s,p}$, with $0 \lt s \lt 1$ and $sp \lt N$, was obtained by Haddad and Ludwig. We generalize their results to the case where $s \gt 1$. 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, Jim\'enez, and Montenegro.