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

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 $U[z,\lambda]$. {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.