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

TL;DR

This paper establishes a linear quantitative stability estimate for the nonlocal Sobolev inequality in the Heisenberg group, relating the distance to optimizers with the functional derivative near solutions, specifically in dimension four.

Contribution

It provides the first linear stability result for the nonlocal Sobolev inequality in the Heisenberg group, especially for weakly interacting bubble solutions in dimension four.

Findings

Linear bound between distance to optimizers and functional derivative.

Stability result holds for solutions close to Euler equation solutions.

Applicable to weakly interacting bubble solutions in dimension four.

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 $Q=4$.