For $ n\geq 2 $, let $ M $ be a $ n $-dimensional connected smooth manifold. We take a fixed smooth measure on $ M $ with strictly positive density, and write $ dx $ as the measure when we integrate. Also, we simply denote the measure of a set $ E\subset M $ by $ |E| $.

Suppose that $ X = (X_{1},X_{2},\cdots,X_{m}) $ are the real $ C^{\infty} $ vector fields defined on $ M $, which satisfy the so-called Hörmander's condition with Hörmander's index $ Q $: there exists a smallest positive integer $ Q $ such that these vector fields $ X_{1},X_{2},\ldots,X_{m} $ together with their commutators of length at most $ Q $ span the tangent space at each point of $ M $.

Starting from the vector fields $ X = (X_{1},X_{2},\cdots,X_{m}) $ which satisfy the Hörmander's condition, we can construct a sub-Riemannian manifold $ (M,D,g) $ endowed with the canonical sub-Riemannian structure $ (D,g) $, where the distribution $ D $ is a family of linear subspaces $ D_{x}\subset T_{x}(M) $ such that $ D_{x} = \rm{span}\{X_{1}(x),X_{2}(x),\cdots,X_{m}(x)\} $ depending smoothly on $ x\in M $, and the sub-Riemannian metric $ g: TM\to \mathbb{R}\cup\{+\infty\} $ is a function given by $ g(x,v) = \inf \left\{\sum\limits_{i = 1}^{m}u_{i}^{2}|v = \sum\limits_{i = 1}^{m}u_{i}X_{i}(x) \right\} $ for $ x\in M $ and $ v\in T_{x}(M) $. Observe that $ g(x,\cdot) $ is a positive definite quadratic form on $ D_{x} $ and $ g(x,v) = +\infty $ for $ v\notin D_{x} $.

For each $ x\in M $, the sub-Riemannian flag at $ x $ is the sequence of nested vector subspaces

$ \{0\} = D_{x}^{0}\subset D_{x} = D_{x}^{1}\subset D_{x}^{2}\subset \cdots \subset D_{x}^{r(x)-1}\subsetneq D_{x}^{r(x)} = T_{x}(M) $

defined in terms of successive Lie brackets, and $ r(x)\leq Q $ is the degree of nonholonomy at $ x $. Here for each $ 1\leq j\leq r(x) $, $ D_{x}^{j} $ is the subspaces of the tangent space at $ x $ spanned by all commutators of $ X_{1},\ldots,X_{m} $ with length at most $ j $. Setting $ \nu_{j}(x) = \dim D_{x}^{j} $ for $ 1\leq j\leq r(x) $ with $ \nu_{0}(x): = 0 $, the integer

$ \begin{equation*} \label{1-1} \nu(x) = \sum\limits_{j = 1}^{r(x)}j(\nu_{j}(x)-\nu_{j-1}(x)) \end{equation*} $

is called the pointwise homogeneous dimension at $ x $. Besides, a point $ x\in M $ is regular if, for every $ 1\leq j\leq r(x) $, the dimension $ \nu_{j}(y) $ is a constant as $ y $ varies in a open neighborhood of $ x $. Otherwise, $ x $ is said to be singular. A set $ S\subset M $ is equiregular if every point of $ S $ is regular. The set $ S\subset M $ is said to be non-equiregular if it contains some singular points. The equiregular assumption in sub-Riemannian geometry is also known as the Métivier's condition in PDEs. For an equiregular connected set $ S $, we know that the pointwise homogeneous dimension $ \nu(x) $ is a constant $ \nu $ which coincides with the Hausdorff dimension of $ S $ related to the vector fields $ X $, and this constant $ \nu $ is also called the Métivier's index. Furthermore, if the set $ S\subset M $ is non-equiregular, we can introduce the so-called generalized Métivier's index by

$ \begin{equation*} \label{1-2} \tilde{\nu}_{S}: = \max\limits_{x\in \overline{S}}\nu(x). \end{equation*} $

The generalized Métivier's index is also known as the non-isotropic dimension of $ S $ related to the vector fields $ X $, which plays a crucial role in the geometry and functional settings associated to the vector fields $ X $. Note that $ n+\max\limits_{x\in \overline{S}}r(x)-1\leq \tilde{\nu}_{S}< nQ $ for $ Q>1 $, and $ \tilde{\nu}_{S} = \nu $ if the closure of $ S $ is equiregular and connected.

In this paper, we concerned with the eigenvalue problems of self-adjoint Hörmander operator $ \triangle_{X}: = -\sum\limits_{i = 1}^{m}X_{i}^{*}X_{i} $ on non-equiregular sub-Riemannian manifolds, where $ X_{i}^{*} $ denotes the formal adjoint of $ X_{i} $. Precisely, we first study the closed eigenvalue problem of $ -\triangle_{X} $, i.e.

$ \begin{equation} -\triangle_{X}u = \mu u\qquad {\rm{in }}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}M{\rm{,}} \end{equation} $

(1.1)

where $ M $ is a $ n $-dimensional connected compact smooth manifold without boundary. Secondly, we study the Dirichlet eigenvalue problem of $ -\triangle_{X} $. For simplicity we assume that $ M $ is an open connected domain in $ \mathbb{R}^{n} $ endowed with Lebesgue measure, and $ \Omega\subset\subset M $ is a bounded connected open subset with smooth boundary $ \partial\Omega $ which is non-characteristic for $ X = (X_{1},X_{2},\cdots,X_{m}) $. The Dirichlet eigenvalue problem of $ -\triangle_{X} $ will be considered as follows

$ \left\{ {\begin{array}{*{20}{l}} { - {\Delta _X}u = \lambda u,}&{{\rm{ in}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \Omega ;}\\ {u = 0,}&{{\rm{ on}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \partial \Omega .} \end{array}} \right. $

(1.2)

In both cases above, the Hörmander's condition ensures that the positive self-adjoint operator $ -\triangle_{X} $ possesses discrete eigenvalues, which will be denoted by $ \{\mu_{k}\}_{k\geq 0} $ and $ \{\lambda_{k}\}_{k\geq 1} $ respectively. Thus we have

$ \begin{eqnarray*} 0& = &\mu_{0}<\mu_1\leq\mu_2\leq\cdots\leq\mu_{k-1}\leq\mu_k\leq\cdots, \\ 0&<&\lambda_1<\lambda_2\leq\cdots\leq\lambda_{k-1}\leq\lambda_k\leq\cdots, \end{eqnarray*} $

and $ \mu_{k}\to +\infty $, $ \lambda_{k}\to +\infty $ as $ k\to +\infty $.

Combining the Rayleigh-Ritz formula with Jerison and Sánchez-Calle's subelliptic heat kernel estimates in [6, 8], we can obtain the following estimate for closed eigenvalue problem (1.1).

Theorem 1.1 Let $ X = (X_{1},X_{2},\cdots,X_{m}) $ be the $ C^{\infty} $ real vector fields defined on the compact manifold $ M $, which satisfy the Hörmander's condition in $ M $. Denote by $ \mu_{k} $ the $ k^{th} $ eigenvalue of the problem (1.1). Then for any $ 0<t<1 $ and any $ k\geq 0 $, we have

$ \begin{equation} \mu_{k+1}\left[A_{1}\int_{M}\frac{dx}{|B_{d_{X}}(x,\sqrt{t})|}-(k+1)\right]+\sum\limits_{j = 0}^{k}\mu_{j}\leq \frac{C_{1,1}}{t}\int_{M}\frac{dx}{|B_{d_{X}}(x,\sqrt{t})|}, \end{equation} $

(1.3)

where $ B_{d_{X}}(x,r) $ denotes the subunit ball induced by the subunit metric, $ A_{1} $ and $ C_{1,1} $ are some positive constants which depend only on the sub-Riemannian structure of $ M $.

Remark 1.1 From Theorem 1.1, we can recover the lower bound estimate of the counting function given by Fefferman and Phong in [4], namely

$ N(\lambda)\geq c_{1}\int_{M}\frac{dx}{|B_{d_{X}}(x,\lambda^{-\frac{1}{2}})|} $

holds for sufficient large $ \lambda>0 $, where $ N(\lambda): = \#\{k\; |\; \mu_{k}\leq \lambda\} $ is the spectral counting function and $ c_{1}>0 $ is a constant depending on the sub-Riemannian structure. This means (1.3) possesses the optimal growth order.

Next, if we denote $ H: = \{x\in M\; |\; \nu(x) = \tilde{\nu} = \max\limits_{x\in M}\nu(x)\} $ as a subset of $ M $. Then the following result gives the explicit upper bound of $ \mu_{k} $.

Theorem 1.2 Suppose that $ X = (X_{1},X_{2},\cdots,X_{m}) $ and $ M $ satisfy the conditions in Theorem 1.1. Denote by $ \tilde{\nu} = \max\limits_{x\in M}\nu(x) $ the non-isotropic dimension of $ M $ related to the vector fields $ X $. If the subset $ H: = \{x\in M\; |\; \nu(x) = \tilde{\nu} = \max\limits_{x\in M}\nu(x)\} $ possesses a positive measure (i.e. $ |H|>0 $), then for any $ k\geq 1 $ we have

$ \begin{equation} \sum\limits_{j = 1}^{k}\mu_{j}\leq \frac{C_{1,1}}{\widehat{C_{1}}}\cdot \left(\frac{\widehat{C_{2}}}{A_{1}}\right)^{1+\frac{2}{\tilde{\nu}}}\cdot \frac{|M|}{|H|^{1+\frac{2}{\tilde{\nu}}}}\cdot (k+1)^{1+\frac{2}{\tilde{\nu}}}, \end{equation} $

(1.4)

and

$ \begin{equation} \mu_{k}\leq \frac{C_{1,1}}{\widehat{C_{1}}}\cdot \left(\frac{2\widehat{C_{2}}}{A_{1}}\right)^{1+\frac{2}{\tilde{\nu}}}\cdot \frac{|M|}{|H|^{1+\frac{2}{\tilde{\nu}}}}\cdot k^{\frac{2}{\tilde{\nu}}}, \end{equation} $

(1.5)

where $ \widehat{C_{1}} $ and $ \widehat{C_{2}} $ are some positive constants depending only on the sub-Riemannian structure of $ M $.

Remark 1.2 From the asymptotic results in [1], we know the upper bounds (1.4) and (1.5) for $ \mu_{k} $ in Theorem 1.2 are optimal in terms of the order on $ k $. In particular, if $ M $ is equiregular, then $ \tilde{\nu} = \nu $ and $ H = M $. In this case, the upper bound estimate (1.5) above gives the similar results by Kokarev [7] and Hassannezhad-Kokarev [5]. However, our results will be suitable for general equiregular sub-Riemannian manifolds and non-equiregular sub-Riemannian manifolds.

Furthermore, we can also obtain the following inequality for the Dirichlet eigenvalues of the problem (1.2).

Theorem 1.3 Let $ X = (X_{1},X_{2},\cdots,X_{m}) $ be $ C^{\infty} $ real vector fields defined on a connected open domain $ M $ in $ \mathbb{R}^n $, which satisfy the Hörmander's condition. Assume that $ \Omega\subset\subset M $ is a bounded connected open subset with smooth boundary such that $ \partial\Omega $ is non-characteristic for $ X $. Denote by $ \lambda_{k} $ the $ k^{th} $ eigenvalue of the problem (1.2). Then for any compact subset $ K\subset \Omega $, there exists a positive constant $ \delta(K) $, such that for any $ 0<t\leq \delta(K) $ and any $ k\geq 1 $, we have

$ \begin{equation} \lambda_{k+1}\left[\frac{A_{2}}{2}\cdot\int_{K} \frac{dx}{|B_{d_{X}}(x,\sqrt{t})|}-k\right]+\sum\limits_{j = 1}^{k}\lambda_{j}\leq \frac{2A_{3}}{t}\int_{\Omega}\frac{1}{|B_{d_{X}}(x,\sqrt{t})|}dx, \end{equation} $

(1.6)

where $ A_{2} $ and $ A_{3} $ are some positive constants which depend only on the sub-Riemannian structure.

Similarly, when the subset $ H $ has a positive measure, Theorem 1.3 also indicates the following explicit upper bounds of $ \lambda_{k} $ which are optimal in terms of the order on $ k $, and also compatible with the asymptotic results in [2].

Theorem 1.4 Suppose that $ X = (X_{1},X_{2},\cdots,X_{m}) $ and $ \Omega $ satisfy the conditions in Theorem 1.3. Let $ \tilde{\nu} = \max\limits_{x\in \overline{\Omega}}\nu(x) $ be the non-isotropic dimension of $ \Omega $ related to the vector fields $ X $. If the subset $ H: = \{x\in \Omega\; |\; \nu(x) = \tilde{\nu} = \max\limits_{x\in \overline{\Omega}}\nu(x)\} $ admits a positive measure, then for any $ k\geq 1 $ we have

$ \begin{equation} \sum\limits_{j = 1}^{k}\lambda_{j}\leq \frac{2A_{3}}{\widehat{C_{1}}}\cdot \left(\frac{4\widehat{C_{2}}}{A_{2}} \right)^{1+\frac{2}{\tilde{\nu}}}\cdot \frac{|\Omega|}{|H|^{1+\frac{2}{\tilde{\nu}}}}\cdot k^{1+\frac{2}{\tilde{\nu}}} \end{equation} $

(1.7)

and

$ \begin{equation} \lambda_{k+1}\leq \frac{2A_{3}}{\widehat{C_{1}}}\cdot \left(\frac{8\widehat{C_{2}}}{A_{2}} \right)^{1+\frac{2}{\tilde{\nu}}}\cdot \frac{|\Omega|}{|H|^{1+\frac{2}{\tilde{\nu}}}}\cdot k^{\frac{2}{\tilde{\nu}}}, \end{equation} $

(1.8)

where $ \widehat{C_{1}} $ and $ \widehat{C_{2}} $ are some positive constants depending only on the sub-Riemannian structure of $ M $.

The details of proofs for Theorem 1.1–Theorem 1.4 have been given in [3].