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Consider the Poincar\'e-Sobolev inequality on the hyperbolic space: for every$n \geq 3$ and $1 < p \leq \frac{n+2}{n-2},$ there exists a best constant$S_{n,p, \lambda}(\mathbb{B}^{n})>0$ such that $$S_{n, p,\lambda}(\mathbb{B}^{n})\left(~\int \limits_{\mathbb{B}^{n}}|u|^{{p+1}} \, {\rmd}v_{\mathbb{B}^n} \right)^{\frac{2}{p+1}}  \leq\int \limits_{\mathbb{B}^{n}}\left(|\nabla_{\mathbb{B}^{n}}u|^{2}-\lambdau^{2}\right) \, {\rm d}v_{\mathbb{B}^n},$$ holds for all $u\inC_c^{\infty}(\mathbb{B}^n),$ and $\lambda \leq \frac{(n-1)^2}{4},$ where$\frac{(n-1)^2}{4}$ is the bottom of the $L^2$-spectrum of$-\Delta_{\mathbb{B}^n}.$ It is known from the results of Mancini and Sandeep[Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008)] that under appropriateassumptions on $n,p$ and $\lambda$ there exists an optimizer, unique up to thehyperbolic isometries, attaining the best constant$S_{n,p,\lambda}(\mathbb{B}^n).$ In this article, we investigate thequantitative gradient stability of the above inequality and the correspondingEuler-Lagrange equation locally around a bubble.  Our result generalizes the sharp quantitative stability of Sobolev inequalityin $\mathbb{R}^n$ of Bianchi-Egnell [J. Funct. Anal. 100 (1991)] andCiraolo-Figalli-Maggi [Int. Math. Res. Not. IMRN 2018] to thePoincar\'{e}-Sobolev inequality on the hyperbolic space.  Furthermore, combining our stability results and implementing a refinedsmoothing estimates, we prove a quantitative extinction rate towards its basinof attraction of the solutions of the sub-critical fast diffusion flow forradial initial data. In another application, we derive sharp quantitativestability of the Hardy-Sobolev-Maz'ya inequalities for the class of functionswhich are symmetric in the component of singularity.

Sharp quantitative stability of Poincare-Sobolev inequality in the hyperbolic space and applications to fast diffusion flows