Degenerate stability of some Sobolev inequalities

Abstract

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.

Rearrangement methods in the work of Elliott Lieb

“…For instance, a quantitative version of the isoperimetric inequality appeared in [55] and quantitative versions of Theorem 13 and Corollary 11 in [36,46]. For further references, we refer to [47].…”

Stability for the Sobolev inequality with explicit constants

“…The proof in [4] proceeds by a spectral estimate combined with a compactness argument and hence cannot give any information about c BE . Explicit quantitative estimates are known only for a distance to M measured by a weaker norm than (2), functions of Ḣ1 (R d ) satisfying additional constraints or superquadratic estimates of the distance which degenerate in a neighbourhood of M and much more is known for subcritical interpolation inequalities than for Sobolev-type inequalities: see [6,26,25,24,7,30] for some references.…”