Stability inequalities with explicit constants for a family of reverse Sobolev inequalities on the sphere

2025/04/28 by König, Tobias · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.19939

Abstract

We prove a stability inequality associated to the reverse Sobolev inequality on the sphere \mathbb Sⁿ, for the full admissible parameter range s - (n)/(2) ∈ (0,1) ∪ (1,2). To implement the classical proof of Bianchi and Egnell, we overcome the main difficulty that the underlying operator A_2s is not positive definite. As a consequence of our analysis and recent results from Gong et al. (arXiv:2503.20350 [math.AP]), the case s - (n)/(2) ∈ (1,2) remarkably constitutes the first example of a Sobolev-type stability inequality (i) whose best constant is explicit and (ii) which does not admit an optimizer.

Citations

• A simple proof of reverse Sobolev inequalities on the sphere and Sobolev trace inequalities on the unit ball
• Stability of the Caffarelli-Kohn-Nirenberg inequality: the existence of minimizers
• On the stability constant of Caffarelli-Kohn-Nirenberg inequality
• Stability for the Sobolev inequality: existence of a minimizer
• Reversed Hardy-Littewood-Sobolev inequality
• Remainder terms in the fractional Sobolev inequality
• A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality
• On the higher order conformal covariant operators on the sphere

Cited by

• An almost-almost-Schur lemma on the 3-sphere
• Sharp stability of the Heisenberg Uncertainty Principle: Second-Order and Curl-Free Field Cases

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