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

Abstract

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 $Q=2n+2$, $n\geq1$, $0<\mu H_{BE}$ 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 $H_{NS}$.