Higher-order affine Sobolev inequalities

2025/06/12 by Tristan Bullion-Gauthier, Bullion-Gauthier, Tristan
Engineering · #Fatigue and fracture mechanics #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2506.10473

Abstract

Zhang refined the classical Sobolev inequality ‖f‖_L^Np/(N-p) \lesssim ‖ ∇ f ‖_L^p, where 1≤ p \lt N, by replacing ‖∇ f‖_L^p with a smaller quantity invariant by unimodular affine transformations. The analogue result in homogeneous fractional Sobolev spaces \mathringW^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, Jiménez, and Montenegro.

Citations

• New approach to affine Moser-Trudinger inequalities via Besov polar projection bodies
• Affine Fractional Sobolev and Isoperimetric Inequalities
• Compactness properties and ground states for the affine Laplacian

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• Affine Hardy--Littlewood--Sobolev inequalities
• Sharp Quantitative Stability for the Affine \(p\)-Sobolev Inequality, Part I: The Case \(2≤ p<n\)
• Affine Logarithmic HLS and Beckner-Type Logarithmic Sobolev Inequalities
• Gagliardo-Nirenberg type inequalities using fractional Sobolev spaces and Besov spaces