Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group

We investigate the optimal Bianchi-Egnell-type quantitative stability constant for the critical nonlocal Sobolev inequality on the Heisenberg group ${\mathbb{H}}^n$, \begin{equation}\label{nS} S_{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}},d\xi d\eta\right)^{\frac{1}{Q^{\ast}{\mu}}} \leq \int{$\mathbb{H}$^{n}}|\nabla_{$\mathbb{H}$}u|^{2}d\xi, \qquad u\in S^{1,2}($\mathbb{H}$^{n}), \end{equation} where $Q=2n+2$, $n\ge 1$, $0<\mu <Q$, and $Q_{\mu }^{\ast }=(2Q-\mu )\mathrm{/}(Q-2)$. Let $\mathfrak{M}$ denote the manifold of Jerison-Lee bubbles, and let $H_{NS}$ be the infimum of the quotient between the deficit in \eqref{nS} and $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(u,\mathfrak{M}{)}^2$. The central issue is that the Euclidean nonlocal problem and the local Folland-Stein-Sobolev problem each possess their own spectral and compactness structures, whereas the present problem couples the HLS interaction with the noncommutative conformal geometry of ${\mathbb{H}}^n$. Under the stated conditions on $(n,\mu )$, these strict thresholds, together with a Heisenberg-group profile decomposition and a nonlocal Br{e}zis-Lieb splitting, imply that $H_{NS}$ is attained. The minimizer then yields the strict comparison $H_{NS}>H_{BE}$ with the optimal stability constant for the local Folland-Stein-Sobolev inequality. We further prove that the sharp universal upper constant for the deficit-to-distance comparison is $1$ and characterize equality. Finally, for the associated Euler-Lagrange equation, we formulate the corresponding residual quotient and derive a strict single-bubble upper bound; this critical-point statement requires a separate expansion and does not follow from attainment of $H_{NS}$.