Optimal Stability of Hardy-Littlewood-Sobolev and Sobolev Inequalities of Arbitrary Orders with Dimension-Dependent Constants

Recently, Dolbeault–Esteban–Figalli–Frank–Loss (Camb J Math 13(2):359–430, 2025) established the optimal stability of the first-order Sobolev inequality with dimension-dependent constant. Subsequently, Chen–Lu–Tang (Adv Math 479:110438, 2025) obtained the optimal stability for the fractional Sobolev inequality of order s when 0