Stability for Critical Points of the Hardy--Littlewood--Sobolev Inequality and a Dual Stability Framework

TL;DR

This paper develops a novel stability framework for critical points of the Hardy--Littlewood--Sobolev inequality, introducing a duality approach and establishing quantitative stability results in non-Hilbertian settings.

Contribution

It introduces a weak-decomposition--strong-stability method and a duality framework, providing the first quantitative stability results for Palais--Smale sequences in a non-Hilbertian context.

Findings

Established a stability inequality for HLS critical points.

Derived explicit lower bounds for Palais--Smale sequence stability.

Extended stability and decomposition results to fractional Sobolev inequalities.

Abstract

Although quantitative stability for critical points of the Sobolev and fractional Sobolev inequalities has been extensively studied, the corresponding stability theory for critical points of the Hardy--Littlewood--Sobolev (HLS) inequality remains largely unexplored. A major difficulty is that the natural stability problem for HLS critical points involves a non-Hilbertian distance, so the classical orthogonal decomposition methods used in Hilbert-space settings are no longer available. In this paper, we develop a weak-decomposition--strong-stability method tailored to the stability structure of HLS critical points and establish the corresponding stability inequality. Our approach also yields an explicit lower bound for the stability of Palais--Smale sequences of the HLS integral equation. To the best of our knowledge, this appears to be the first quantitative stability result for Palais--Smale sequences of a variational functional measured in a non-Hilbertian distance. We further introduce a duality framework connecting Struwe-type decompositions and stability inequalities for critical points of the Sobolev inequality with their HLS counterparts. As a consequence, we derive Struwe-type decomposition and stability results for critical points of the fractional Sobolev inequality for general functions, thereby removing the nonnegativity assumption imposed in [26].