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arXiv · 2207.06375

Affine Fractional Sobolev and Isoperimetric Inequalities

Julián Haddad·Monika Ludwig
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Abstract

Sharp affine fractional Sobolev inequalities for functions on $\mathbb R^n$ are established. For each $0<s<1$, the new inequalities are significantly stronger than (and directly imply) the sharp fractional Sobolev inequalities of Almgren and Lieb. In the limit as $s\to 1^-$, the new inequalities imply the sharp affine Sobolev inequality of Gaoyong Zhang. As a consequence, fractional Petty projection inequalities are obtained that are stronger than the fractional Euclidean isoperimetric inequalities and a natural conjecture for radial mean bodies is proved.

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Julián Haddad, Monika Ludwig. 2022-07-13. Affine Fractional Sobolev and Isoperimetric Inequalities. https://doi.org/10.4310/jdg/1740137450

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