Geodesic
Degenerate stability of some Sobolev inequalities
Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 6, pp. 1459-1484

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We show that on 𝕊 1 (1/d-2)×𝕊 d-1 (1) the conformally invariant Sobolev inequality holds with a remainder term that is the fourth power of the distance to the optimizers. The fourth power is best possible. This is in contrast to the more usual vanishing to second order and is motivated by work of Engelstein, Neumayer and Spolaor. A similar phenomenon arises for subcritical Sobolev inequalities on 𝕊 d . Our proof proceeds by an iterated Bianchi–Egnell strategy.

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DOI : 10.4171/aihpc/35
Classification : 39B62, 49K40, 35B35
Keywords: remainder term, Sobolev inequality, stability 
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     author = {Frank, Rupert L.},
     title = {Degenerate stability of some {Sobolev} inequalities},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1459--1484},
     volume = {39},
     number = {6},
     year = {2022},
     doi = {10.4171/aihpc/35},
     language = {en},
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Frank, Rupert L. Degenerate stability of some Sobolev inequalities. Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 6, pp. 1459-1484. doi: 10.4171/aihpc/35

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