Stability of Hardy Littlewood Sobolev inequality under bubbling

Abstract In this note we will generalize the results deduced in Figalli and Glaudo (Arch Ration Mech Anal 237 (1): 201–258, 2020) and Deng et al. (Sharp quantitative estimates of Struwe’s Decomposition. Preprint http: //arxiv. org/abs/2103. 15360, 2021) to fractional Sobolev spaces. In particular we will show that for s (0, 1) s ∈ (0, 1), n>2s n > 2 s and N ν ∈ N there exists constants = (n, s, ) >0 δ = δ (n, s, ν) > 0 and C=C (n, s, ) >0 C = C (n, s, ν) > 0 such that for any function u Ḣˢ (Rⁿ) u ∈ H ˙ s (R n) satisfying, aligned \| u- ₈=₁^ U₈\| ₇̇⌁ aligned u - ∑ i = 1 ν U ~ i H ˙ s ≤ δ where U₁, U₂, U U ~ 1, U ~ 2, … U ~ ν is a δ -interacting family of Talenti bubbles, there exists a family of Talenti bubbles U₁, U₂, U U 1, U 2, … U ν such that aligned \| u- ₈=₁^ U₈\| ₇̇⌁ C\ array{ll & if 2s 6s array. aligned u - ∑ i