Stability of Hardy Littlewood Sobolev Inequality under Bubbling

TL;DR

This paper extends the stability analysis of the Hardy-Littlewood-Sobolev inequality to fractional Sobolev spaces, showing how functions close to Talenti bubbles can be approximated by exact bubbles with quantifiable error bounds.

Contribution

It generalizes previous results to fractional Sobolev spaces, establishing stability estimates for solutions near Talenti bubbles in these spaces.

Findings

Stability estimates depend on the dimension and fractional order.

Error bounds involve the norm of the residual in the dual Sobolev space.

Results apply to functions close to a family of Talenti bubbles in fractional Sobolev norms.

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}^{\nu} \tilde{U}_{i}\right\|_{\dot{H}^s} \leq \delta \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}^{\nu} U_{i}\right\|_{\dot{H}^s} \leq 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{align*} for $\Gamma=\left\|\Delta u+u|u|^{p-1}\right\|_{H^{-s}}$ and $p=2^*-1=\frac{n+2s}{n-2s}.$