Stability for Strichartz inequalities: Existence of minimizers

TL;DR

This work analyzes quantitative stability of Strichartz inequalities for the paraboloid and the two-dimensional sphere, focusing on the existence of minimizers for the sharp stability constants. It derives explicit spectral-gap constants and two-peak thresholds, showing the two-peak obstruction vanishes in both settings and that minimizers exist under a strict paraboloid condition $C_{*} < \widetilde{C}_{d}$, while the sphere case yields unconditional minimizer existence since $C_{**} < C_{SG*}$. The proofs combine concentration-compactness/profile decomposition with detailed spectral analysis around Gaussian and constant maximizers, yielding precise constants such as $C_{SG}=\widetilde{C}_d$ and $C_{SG*}=8\pi^2/5$. These results illuminate stability thresholds for Strichartz inequalities and reveal distinct behaviors between paraboloid and sphere cases at the level of minimizers and compactness phenomena.

Abstract

We study the quantitative stability associated with the adjoint Fourier restriction inequality, focusing on the paraboloid and two-dimensional sphere cases. We show that these Strichartz-stability inequalities admit minimizers attaining their sharp constants, provided that these sharp constants are strictly smaller than the corresponding spectral-gap constants. Furthermore, for the two-dimensional sphere case, we obtain the existence of minimizers.

Theorems & Definitions (23)

• Theorem 1.1: Sphere-stability minimizer
• Remark 1.2
• Theorem 1.3: Paraboloid-stability minimizer
• Remark 1.4
• Conjecture 1.5: Gaussian maximizer
• Proposition 1.6: Paraboloid spectral-gap constant
• Proposition 1.7: Paraboloid two-peak constant
• Proposition 1.8: Paraboloid two-peak vanishing
• Remark 1.9
• proof : Proof of Proposition \ref{['P:Paraboloid two-peak vanishing']}