Let $\mathrm{\Omega }$ be a bounded connected open subset in ${\mathbb{R}}^n$ with smooth boundary $\partial \mathrm{\Omega }$. Suppose that we have a system of real smooth vector fields $X=(X_1,X_2$, $\dots ,X_m)$ defined on a neighborhood of $\overline{\mathrm{\Omega }}$ that satisfies the Hörmander's condition. Suppose further that $\partial \mathrm{\Omega }$ is non-characteristic with respect to $X$. For a self-adjoint sub-elliptic operator ${▵}_X=-{\sum }_{i=1}^m{X}_i^{\ast }{X}_i$ on $\mathrm{\Omega }$, we denote its $k\text{th}$ Dirichlet eigenvalue by ${\lambda }_k$. We will provide a uniform upper bound for the sub-elliptic Dirichlet heat kernel. We will also give an explicit sharp lower bound estimate for ${\lambda }_k$, which has a polynomially growth in $k$ of the order related to the generalized Métivier index. We will establish an explicit asymptotic formula of ${\lambda }_k$ that generalizes the Métivier's results in 1976. Our asymptotic formula shows that under a certain condition, our lower bound estimate for ${\lambda }_k$ is optimal in terms of the growth of $k$. Moreover, the upper bound estimate of the Dirichlet eigenvalues for general sub-elliptic operators will also be given, which, in a certain sense, has the optimal growth order.

中文翻译：

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一类亚椭圆算子的狄利克雷特征值估计

让 $\mathrm{\Omega }$ 是一个有界连通开子集 ${\mathbb{电阻}}^n$ 边界平滑 $\partial \mathrm{\Omega }$. 假设我们有一个实数平滑向量场系统$X=(X_1,X_2$, $\dots ,X_{米})$ 定义在一个邻域上 $\overline{\mathrm{\Omega }}$满足荷曼德条件。进一步假设$\partial \mathrm{\Omega }$ 是非特征性的 $X$. 对于自伴随子椭圆算子${▵}_X=-{\sum }_{\mathrm{一世}=1}^{米}{X}_{\mathrm{一世}}^{＊}{X}_{\mathrm{一世}}$ 上 $\mathrm{\Omega }$，我们表示其 $克\text{日}$ 狄利克雷特征值由 ${\lambda }_{克}$. 我们将为亚椭圆 Dirichlet 热核提供统一的上限。我们还将给出一个明确的尖锐下界估计${\lambda }_{克}$, 有多项式增长 $克$与广义 Métivier 指数相关的顺序。我们将建立一个显式渐近公式${\lambda }_{克}$ 概括了 Métivier 在 1976 年的结果。我们的渐近公式表明，在一定条件下，我们对下界的估计 ${\lambda }_{克}$ 就生长而言是最优的 $克$. 此外，还将给出一般亚椭圆算子的狄利克雷特征值的上界估计，它在某种意义上具有最优增长阶数。