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

We extend the affine inequalities on ${\mathbb{R}}^n$ for Sobolev functions in $W^{s,p}$ with $1\le p<n\mathrm{/}s$ obtained recently by Haddad-Ludwig [16, 17] to the remaining range $p\ge n\mathrm{/}s$. 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.