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,μ) \left(\int_{\mathbb{H}^{n}}\int_{\mathbb{H}^{n}}\frac{|u(ξ)|^{Q^{\ast}_μ}|u(η)|^{Q^{\ast}_μ}}{|η^{-1}ξ|^μ}\mathrm{d}ξ\mathrm{d}η\right)^{\frac{1}{Q^{\ast}_μ}}\leq \int_{\mathbb{H}^{n}}|\nabla_{H}u|^{2}dξ,\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 $Q_{\mu }^{\ast }=\frac{2Q-\mu }{Q-2}$ 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} -Δ_{H} u=\left(\int_{\mathbb{H}^{n}}\frac{|u(η)|^{Q^{\ast}_μ}}{|η^{-1}ξ|^μ}\mathrm{d}η\right)|u|^{Q^{\ast}_μ-2}u,\qquadξ,η\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*} Γ(u)=\left\|Δ_{H}u+\left(\int_{\mathbb{H}^{n}}\frac{|u(η)|^{Q^{\ast}_μ}}{|η^{-1}ξ|^μ}\mathrm{d}η\right)|u|^{Q^{\ast}_μ-2}u\right\|_{(S^{1,2}(\mathbb{H}^{n}))^{-1}}. \end{equation*} And for the weakly interacting bubble solutions $\sum _{i=1}^{\nu }{U}_{{\lambda }_i,{\zeta }_i}$, the aforementioned quantitative stability result holds when the dimension $Q=4$.

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