海森堡群上临界非局部索伯列夫不等式的最优稳定性界、极小元与临界点

中文摘要 AI 辅助

我们研究海森堡群$\boldsymbol{\reals^n}$上临界非局部索伯列夫不等式的最优Bianchi-Egnell型定量稳定性常数，其不等式形式为：$S_{HL}(Q,\boldsymbol{\reals^n})\bigg(\textstyle\bigtimes\boldsymbol{\reals^n}\textstyle\bigtimes\boldsymbol{\reals^n}\frac{|u(\boldsymbol{\reals})|^{Q^{\boldsymbol{\reals}}_{\boldsymbol{\reals}}}|u(\boldsymbol{\reals})|^{Q^{\boldsymbol{\reals}}_{\boldsymbol{\reals}}}}{|\boldsymbol{\reals}^{-1}\boldsymbol{\reals}|^{\boldsymbol{\reals}}}\boldsymbol{d}\boldsymbol{\reals}\boldsymbol{d}\boldsymbol{\reals}\bigg)^{\frac{1}{Q^{\boldsymbol{\reals}}_{\boldsymbol{\reals}}}} \textstyle\bigtimes\boldsymbol{\reals^n}|\nabla_{\boldsymbol{\reals}}u|^{2}\boldsymbol{d}\boldsymbol{\reals}$，其中$Q=2n+2$，$n\boldsymbol{\reals}1$，$0\boldsymbol{\reals}\boldsymbol{\reals}Q$，$Q^{\boldsymbol{\reals}}_{\boldsymbol{\reals}}=(2Q-\boldsymbol{\reals})/(Q-2)$。记$\boldsymbol{\reals}$为Jerison-Lee泡流形，$H_{NS}$为式(nS)中亏缺与$\boldsymbol{\reals}(u,\boldsymbol{\reals})^2$之比的下确界。核心问题在于：欧氏非局部问题与局部Folland-Stein-Sobolev问题各有其谱与紧性结构，而本问题将HLS相互作用与$\boldsymbol{\reals^n}$的非交换共形几何耦合。在$(n,\boldsymbol{\reals})$的给定条件下，这些严格阈值结合海森堡群剖面分解与非局部Br{e}zis-Lieb分裂，可推出$H_{NS}$可达。极小元进而给出严格比较$H_{NS}>H_{BE}$（$H_{BE}$为局部Folland-Stein-Sobolev不等式的最优稳定性常数）。我们还证明亏缺-距离比较的尖锐通用上常数为1，并刻画等号成立条件。最后，对关联欧拉-拉格朗日方程，我们构造对应残差商并推导严格单泡上界；该临界点结论需单独展开，无法由$H_{NS}$的可达性直接推出。

英文摘要

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,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}\,dξdη\right)^{\frac{1}{Q^{\ast}_μ}} \leq \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}}u|^{2}dξ, \qquad u\in S^{1,2}(\mathbb{H}^{n}), \end{equation} where $Q=2n+2$, $n\geq1$, $0<μ<Q$, and $Q^{\ast}_μ=(2Q-μ)/(Q-2)$. 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 $\mathrm{dist}(u,\mathfrak{M})^{2}$. 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 $(n,μ)$, 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 $H_{NS}$.

发表机构

• School of Mathematics and Statistics, Southwest University（西南大学数学与统计学院）

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