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

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].