Quantitative stability of critical points for the nonlocal-Sobolev inequality in Heisenberg group

Abstract

We investigate the quantitative stability of the nonlocal Sobolev inequality in Heisenberg group \begin{equation*}\label{non-Sobolev} C_{HL}(Q,\mu) \left(\int_{\mathbb{H}^{n}}\int_{\mathbb{H}^{n}}\frac{|u(\xi)|^{Q^{\ast}_{\mu}}|u(\eta)|^{Q^{\ast}_{\mu}}}{|\eta^{-1}\xi|^{\mu}}\mathrm{d}\xi\mathrm{d}\eta\right)^{\frac{1}{Q^{\ast}_{\mu}}}\leq \int_{\mathbb{H}^{n}}|\nabla_{H}u|^{2}d\xi,\qquad\forall u\in S^{1,2}(\mathbb{H}^{n}), \end{equation*} where $Q=2n+2$ is the homogeneous dimension of the Hiesenberg group $\mathbb{H}^{n}$, $\mu\in(0,Q)$ and $Q^{\ast}_{\mu}=\frac{2Q-\mu}{Q-2}$ are two parameters corresponding to the Hardy-Littlewood-Sobolev inequality and Folland-Stein inequality on Heisenberg group, $C_{HL}(Q,\mu)$ is the sharp constant of the nonlocal-Sobolev inequality. Specifically, when $u$ is close to solving the Euler equation \begin{equation*}\label{non-critical-n} -\Delta_{H} u=\left(\int_{\mathbb{H}^{n}}\frac{|u(\eta)|^{Q^{\ast}_{\mu}}}{|\eta^{-1}\xi|^{\mu}}\mathrm{d}\eta\right)|u|^{Q^{\ast}_{\mu}-2}u,\qquad\xi,\eta\in\mathbb{H}^{n}, \end{equation*} the natural distance between $u$ and the the set of optimizers $U_{\lambda,\zeta}$, defined as $\delta(u)=||\nabla_{H}u-\nabla_{H}U_{\lambda,\zeta}||_{L^{2}}$, can be linearly bounded by the functional derivative term \begin{equation*} \Gamma(u)=\left\|\Delta_{H}u+\left(\int_{\mathbb{H}^{n}}\frac{|u(\eta)|^{Q^{\ast}_{\mu}}}{|\eta^{-1}\xi|^{\mu}}\mathrm{d}\eta\right)|u|^{Q^{\ast}_{\mu}-2}u\right\|_{(S^{1,2}(\mathbb{H}^{n}))^{-1}}. \end{equation*} And for the weakly interacting bubble solutions $\mathop{\sum}\limits_{i=1}^{\nu}U_{\lambda_{i},\zeta_{i}}$, the aforementioned quantitative stability result holds when the dimension $Q=4$.