Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion

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 $$\frac{1}{2}{S}_{N,s}^{\frac{N}{2s}}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{{\dot{H}}^s({\mathbb{R}}^N)}\le \frac{3}{2}{S}_{N,s}^{\frac{N}{2s}},$$ where $S_{N,s}$ is the optimal Sobolev constant, the bound $$\mathrm{\parallel }u-U[z,\lambda ]{\mathrm{\parallel }}_{{\dot{H}}^s({\mathbb{R}}^N)}\lesssim \mathrm{\parallel }(-\mathrm{\Delta }{)}^{s}u-u^{2_s^{\ast }-1}{\mathrm{\parallel }}_{{\dot{H}}^{-s}({\mathbb{R}}^N)},$$ holds for a suitable fractional Talenti bubble $U[z,\lambda ]$. {For functions $u$ which are close to Talenti bubbles, we give the sharp asymptotic value of the implied constant in this inequality.} As an application {of this}, we derive an explicit polynomial extinction rate for positive solutions to a fractional fast diffusion equation.