We extend the affine inequalities on Rⁿ for Sobolev functions in Ws,p with 1 ≤ p < n/s obtained recently by Haddad-Ludwig [16, 17] to the remaining range p ≥ 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 Lᵖ 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.
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Domínguez et al. (2024) studied this question.
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