Stability of Hardy Littlewood Sobolev Inequality under Bubbling

Abstract

In this note we will generalize the results deduced in arXiv:1905.08203 and arXiv:2103.15360 to fractional Sobolev spaces. In particular we will show that for $s\in (0,1)$, $n>2s$ and $\nu \in \mathbb{N}$ there exists constants $\delta =\delta (n,s,\nu )>0$ and $C=C(n,s,\nu )>0$ such that for any function $u\in {\dot{H}}^s({\mathbb{R}}^n)$ satisfying, \begin{align*} \left\| u-\sum_{i=1}^ν \tilde{U}_{i}\right\|_{\dot{H}^s} \leq δ\end{align*} where ${\tilde{U}}_1,{\tilde{U}}_2,\cdots {\tilde{U}}_{\nu }$ is a $\delta -$interacting family of Talenti bubbles, there exists a family of Talenti bubbles $U_1,U_2,\cdots U_{\nu }$ such that \begin{align*} \left\| u-\sum_{i=1}^ν U_{i}\right\|_{\dot{H}^s} \leq C\left\{\begin{array}{ll} Γ& \text { if } 2s < n < 6s,\\ Γ|\log Γ|^{\frac{1}{2}} & \text { if } n=6s, \\ Γ^{\frac{p}{2}} & \text { if } n > 6s \end{array}\right. \end{align*} for $\Gamma ={\Vert \Delta u+u|u{|}^{p-1}\Vert }_{H^{-s}}$ and $p=2^{\ast }-1=\frac{n+2s}{n-2s}.$

arXiv comment38 pages, to appear in Calc. Var. and PDE