New approach to affine Moser-Trudinger inequalities via Besov polar projection bodies

TL;DR

This paper extends affine Sobolev inequalities to a broader range of parameters using Besov polar projection bodies, strengthening known inequalities and establishing new analogs of classical results.

Contribution

It generalizes affine Moser-Trudinger and Morrey inequalities for Sobolev functions to the full range p ≥ n/s using Besov polar projection bodies.

Findings

Stronger affine Moser-Trudinger and Morrey inequalities for all p ≥ n/s.

Establishment of an analog of Bourgain-Brezis-Mironescu inequalities for p=n.

Introduction of Besov polar projection bodies as a key tool.

Abstract

We extend the affine inequalities on $\mathbb{R}^n$ for Sobolev functions in $W^{s,p}$ with $1 \leq p < n/s$ obtained recently by Haddad-Ludwig [16, 17] to the remaining range $p \geq n/s$. For each value of $s$, our results are stronger than affine Moser-Trudinger and Morrey inequalities. As a byproduct, we establish the analog of the classical $L^p$ Bourgain-Brezis-Mironescu inequalities related to the Moser-Trudinger case $p=n$. Our main tool is the affine invariant provided by Besov polar projection bodies.