*Submitted Paper*

**Inserted:** 19 nov 2022

**Last Updated:** 22 nov 2022

**Year:** 2022

**Abstract:**

We study the quantitative stability of critical points of the fractional Sobolev inequality. We show that, for a non-negative function $u \in \dot H^s(\mathbb R^N)$ whose energy satisfies
\[ \tfrac{1}{2} S^\frac{N}{2s}_{N,s} \le \

u\

_{\dot H^s(\mathbb R^N)} \le \tfrac{3}{2}S_{N,s}^\frac{N}{2s},
\]
where $S_{N,s}$ is the optimal Sobolev constant, the bound
\[ \

u -U[z,\lambda]\

_{\dot{H}^s(\mathbb R^N)} \lesssim \

(-\Delta)^s u - u^{2^*_s-1}\

_{\dot{H}^{-s}(\mathbb R^N)},
\]
holds for a suitable fractional Talenti bubble $U[z,\lambda]$. Special attention is paid to the explicit value of the implied constant in this stability inequality, which is new even for $s = 1$. As an application, we derive an explicit polynomial extinction rate for positive solutions to a fractional fast diffusion equation.