Stability of Hardy Littlewood Sobolev inequality under bubbling

Abstract

AbstractIn 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 $$s\in (0,1)$$ s ∈ ( 0 , 1 ) , $$n>2s$$ n > 2 s and $$\nu \in \mathbb {N}$$ ν ∈ N there exists constants $$\delta = \delta (n,s,\nu )>0$$ δ = δ ( n , s , ν ) > 0 and $$C=C(n,s,\nu )>0$$ C = C ( n , s , ν ) > 0 such that for any function $$u\in \dot{H}^s(\mathbb {R}^n)$$ u ∈ H ˙ s ( R n ) satisfying, $$\begin{aligned} \left\| u-\sum _{i=1}^{\nu } \tilde{U}_{i}\right\| _{\dot{H}^s} \le \delta \end{aligned}$$ u - ∑ i = 1 ν U ~ i H ˙ s ≤ δ where $$\tilde{U}_{1}, \tilde{U}_{2},\ldots \tilde{U}_{\nu }$$ U ~ 1 , U ~ 2 , … U ~ ν is a $$\delta $$ δ -interacting family of Talenti bubbles, there exists a family of Talenti bubbles $$U_{1}, U_{2},\ldots U_{\nu }$$ U 1 , U 2 , … U ν such that $$\begin{aligned} \left\| u-\sum _{i=1}^{\nu } U_{i}\right\| _{\dot{H}^s} \le C\left\{ \begin{array}{ll} \Gamma &{} \text{ if } 2s< n < 6s,\\ \Gamma |\log \Gamma |^{\frac{1}{2}} &{} \text{ if } n=6s, \\ \Gamma ^{\frac{p}{2}} &{} \text{ if } n > 6s \end{array}\right. \end{aligned}$$ u - ∑ i = 1 ν U i H ˙ s ≤ C Γ if 2 s < n < 6 s , Γ | log Γ | 1 2 if n = 6 s , Γ p 2 if n > 6 s for $$\Gamma =\left\| \Delta u+u|u|^{p-1}\right\| _{H^{-s}}$$ Γ = Δ u + u | u | p - 1 H - s and $$p=2^*-1=\frac{n+2s}{n-2s}.$$ p = 2 ∗ - 1 = n + 2 s n - 2 s .