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Mathematics > Metric Geometry

arXiv:2207.06375 (math)
[Submitted on 13 Jul 2022 (v1), last revised 22 Dec 2022 (this version, v2)]

Title:Affine Fractional Sobolev and Isoperimetric Inequalities

Authors: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.
Subjects: Metric Geometry (math.MG); Functional Analysis (math.FA)
MSC classes: 2020: 46E35 (Primary) 35R11, 52A40 (Secondary)
Cite as: arXiv:2207.06375 [math.MG]
  (or arXiv:2207.06375v2 [math.MG] for this version)
  https://doi.org/10.48550/arXiv.2207.06375

Submission history

From: Julián Haddad [view email]
[v1] Wed, 13 Jul 2022 17:29:29 UTC (17 KB)
[v2] Thu, 22 Dec 2022 10:29:41 UTC (17 KB)
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