Estimates of Dirichlet Eigenvalues for a Class of Sub-elliptic Operators

TL;DR

This paper derives explicit bounds and asymptotic formulas for Dirichlet eigenvalues of a class of sub-elliptic operators satisfying Hörmander's condition, extending classical results and providing optimal growth estimates.

Contribution

It provides new explicit bounds and asymptotic formulas for eigenvalues of sub-elliptic operators, generalizing Métivier's classical results from 1976.

Findings

Established uniform upper bounds for the sub-elliptic Dirichlet heat kernel.

Derived explicit sharp lower bounds for eigenvalues with polynomial growth in k.

Presented an asymptotic formula for eigenvalues that generalizes and extends classical results.

Abstract

Let $\Omega$ be a bounded connected open subset in $\mathbb{R}^n$ with smooth boundary $\partial\Omega$. Suppose that we have a system of real smooth vector fields $X=(X_{1},X_{2},$ $\cdots,X_{m})$ defined on a neighborhood of $\overline{\Omega}$ that satisfies the H\"{o}rmander's condition. Suppose further that $\partial\Omega$ is non-characteristic with respect to $X$. For a self-adjoint sub-elliptic operator $\triangle_{X}= -\sum_{i=1}^{m}X_{i}^{*} X_i$ on $\Omega$, we denote its $k^{th}$ Dirichlet eigenvalue by $\lambda_k$. We will provide an 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 M\'{e}tivier index. We will establish an explicit asymptotic formula of $\lambda_{k}$ that generalizes the M\'{e}tivier'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.

Equations

$$H_X^1(W)=\{u\in L^2(W)|X_{j}u\in L^2(W),j=1,\cdots ,m\}.$$

$${△}_X:=-\sum _{i=1}^m{X}_i^{\ast }{X}_i,$$

$$\left\{\begin{array}[]{ll}-\triangle_{X}u=\lambda u,&\hbox{in $\Omega$;}\\[8.53581pt] u=0,&\hbox{on $\partial\Omega$.}\end{array}\right.$$

$${\lambda }_k\sim {\left({\int }_{\Omega }\gamma (x)dx\right)}^{-\frac{2}{\nu }}\cdot k^{\frac{2}{\nu }}\text{mbox}ask\to +\infty ,$$

$$\nu :=\sum _{j=1}^{Q}j({\nu }_j-{\nu }_{j-1}),{\nu }_0:=0,$$

$$\sum _{j=1}^k{\lambda }_j\ge C_0\cdot k^{1+\frac{2}{nQ}}\text{mbox}foranyk\ge 1,$$

$$\nu (x):=\sum _{j=1}^{Q}j({\nu }_j(x)-{\nu }_{j-1}(x)),{\nu }_0(x):=0.$$

$$\tilde{\nu }:=\underset{x\in \overline{\Omega }}{\max }\nu (x)$$

$$h_D(x,x,t)\le \frac{C}{t^{\frac{\tilde{\nu }}{2}}}\text{mbox}forall(x,t)\in \Omega \times (0,+\infty ),$$

$$\sum _{j=1}^k{\lambda }_j\ge C_1\cdot k^{1+\frac{2}{\tilde{\nu }}},$$

$$\underset{\lambda \to +\infty }{\lim }{\lambda }^{-\frac{\tilde{\nu }}{2}}N(\lambda )=\frac{1}{\Gamma \left(\frac{\tilde{\nu }}{2}+1\right)}\cdot {\int }_H{\gamma }_0(x)dx.$$

$${\lambda }_k={\left(\frac{\Gamma \left(\frac{\tilde{\nu }}{2}+1\right)}{{\int }_H{\gamma }_0(x)dx}\right)}^{\frac{2}{\tilde{\nu }}}\cdot k^{\frac{2}{\tilde{\nu }}}+o(k^{\frac{2}{\tilde{\nu }}})\text{mbox}ask\to +\infty ;$$

$$\underset{k\to +\infty }{\lim }\frac{k^{\frac{2}{\tilde{\nu }}}}{{\lambda }_k}=0.$$

$${\lambda }_k\le \tilde{C}\cdot (k-1{)}^{\frac{2}{n}}+{\lambda }_1\text{mbox}forallk\ge 1,$$

$${\int }_{\Omega }\frac{dx}{\sum |\det (Y_{i_1},Y_{i_2},\cdots ,Y_{i_n})(x)|}<+\infty .$$

$$\sum _{i=1}^k{\lambda }_i\ge C\cdot k^{1+\frac{2}{n}}\text{mbox}forallk\ge 1.$$

$$\Vert u{\Vert }_{L^q(\Omega )}\le C\left(\Vert Xu{\Vert }_{L^p(\Omega )}+\Vert u{\Vert }_{L^p(\Omega )}\right)$$

$${\left({\int }_{\Omega }|u{|}^{\frac{2\tilde{\nu }}{\tilde{\nu }-2}}dx\right)}^{\frac{\tilde{\nu }-2}{2\tilde{\nu }}}\le C{\left({\int }_{\Omega }|Xu{|}^{2}dx+{\int }_{\Omega }|u{|}^{2}dx\right)}^{\frac{1}{2}},$$

$${\lambda }_1{\int }_{\Omega }|u{|}^{2}dx\le {\int }_{\Omega }|Xu{|}^{2}dx,\forall u\in H_{X,0}^1(\Omega ).$$

$${\lambda }_1=\underset{\Vert \phi {\Vert }_2=1,\phi \in H_{X,0}^1(\Omega )}{\inf }\Vert X\phi {\Vert }_{L^2(\Omega )}^2.$$

$${\left({\int }_{\Omega }|u{|}^{\frac{2\tilde{\nu }}{\tilde{\nu }-2}}dx\right)}^{\frac{\tilde{\nu }-2}{2\tilde{\nu }}}\le C{\left({\int }_{\Omega }|Xu{|}^{2}dx\right)}^{\frac{1}{2}}.$$

$$\Vert u{\Vert }_{H^{{ϵ}_0}({\mathbb{R}}^n)}^2\le C\left(\sum _{i=1}^m\Vert X_{i}u{\Vert }_{L^2({\mathbb{R}}^n)}^2+\Vert u{\Vert }_{L^2({\mathbb{R}}^n)}^2\right),\forall u\in H_{X,0}^1(\Omega ).$$

$$\Vert \varphi u{\Vert }_{H^{s+ϵ}({\mathbb{R}}^n)}\le C\left(\Vert {\varphi }_1{△}_{X}u{\Vert }_{H^s({\mathbb{R}}^n)}+\Vert {\varphi }_{1}u{\Vert }_{L^2({\mathbb{R}}^n)}\right),$$

$$\underset{x\in {\mathbb{R}}^n}{\sup }|u(x)|\le C\Vert u{\Vert }_{H^s({\mathbb{R}}^n)}\text{mbox}forallu\in H^s({\mathbb{R}}^n).$$

$$\underset{x\in \Omega }{\sup }|\xi (x)u(x)|\le C\sum _{k=0}^N\Vert ({△}_X{)}^{k}u{\Vert }_{L^2(\Omega )}.$$

$${\phi }^{\prime }(t)=\sum _{i=1}^m{a}_i(t)X_i(\phi (t))\text{mbox}a.eforallt\in [0,1]$$

$$d_X(x,y):=\inf \{\delta >0|\exists \phi \in C(x,y,\delta )\text{mbox}with\phi (0)=x,\phi (1)=y\}.$$

$$B_{d_X}(x,r):=\{y\in V|d_X(x,y)<r\}$$

$$B(x,r)\subset B_{d_X}(x,c{r}^{{ϵ}_0}),$$

$$X_I=[X_{j_1},[X_{j_2},\cdots [X_{j_{k-1}},X_{j_k}]\cdots ]].$$

Full text

††journal: Elsevier

1 Introduction and Main Results

For $n\geq 2$, let $X=(X_{1},X_{2},\cdots,X_{m})$ be the system of $C^{\infty}$ real vector fields defined over a domain $W$ in $\mathbb{R}^{n}$. For our study here, the essential hypothesis is the following Hörmander’s condition: (cf. [22])

(H): $X_{1},X_{2},\ldots,X_{m}$ together with their commutators up to a certain fixed length span the tangent space at each point of $W$.

We introduce the following weighted Sobolev spaces (cf. [49]) associated with $X$,

[TABLE]

Then $H_{X}^{1}(W)$ is a Hilbert space endowed with norm $\|u\|^{2}_{H^{1}_{X}(W)}=\|u\|_{L^{2}(W)}^{2}+\|Xu\|_{L^{2}(W)}^{2}$, where $\|Xu\|_{L^{2}(W)}^{2}=\sum_{j=1}^{m}\|X_{j}u\|_{L^{2}(W)}^{2}$.

Let $\Omega\subset\subset W$ be a bounded connected open subset with $C^{\infty}$ boundary and the boundary $\partial\Omega$ is assumed to be non-characteristic for $X$ (i.e. for any $x_{0}\in\partial\Omega$, there exists at least one vector field $X_{j_{0}}~{}(1\leq j_{0}\leq m)$, such that $X_{j_{0}}(x_{0})\notin T_{x_{0}}(\partial\Omega)$). Then the space $H_{X,0}^{1}(\Omega)$ being the closure of $C_{0}^{\infty}(\Omega)$ in $H_{X}^{1}(W)$ is well-defined, and is also a Hilbert space. Clearly, the vector fields in $X$ satisfy the condition (H) on $\overline{\Omega}$. Hence there is an integer $Q$ such that the vector fields $X_{1},X_{2},\ldots,X_{m}$ together with their commutators of length at most $Q$ span the tangent space $T_{x}(W)$ at each point $x\in\overline{\Omega}$. Recall that $Q$ is called the Hörmander’s index of $\overline{\Omega}$ with respect to $X$. We say that the vector fields $X$ are finitely degenerate if $2\leq Q<+\infty$.

Consider the following Hörmander type operator

[TABLE]

where $X_{i}^{*}$ is the formal adjoint of $X_{i}$. (In general, $X_{i}^{*}=-X_{i}-\text{div}X_{i}$, where $\text{div}X_{i}$ is the divergence of $X_{i}$.) Since $-\triangle_{X}$ is symmetric on $C_{0}^{\infty}(\Omega)$, it is easy to show that, after self-adjoint extension, $-\triangle_{X}$ can be uniquely extended to a positive unbounded self-adjoint operator on the domain $D(\triangle_{X})=\{u\in H_{X,0}^{1}(\Omega)|\triangle_{X}u\in L^{2}(\Omega)\}$.

In this paper, we mainly focus on the following Dirichlet eigenvalue problem in $H_{X,0}^{1}(\Omega)$,

[TABLE]

From the condition (H), we know that the sub-elliptic self-adjoint operator $-\triangle_{X}$ defined on $D(\triangle_{X})$ has discrete eigenvalues $0<\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{k-1}\leq\lambda_{k}\leq\cdots$, and $\lambda_{k}\to+\infty$ as $k\to+\infty$.

When $X=(\partial_{x_{1}},\cdots,\partial_{x_{n}})$, $\triangle_{X}$ is the standard Laplacian $\triangle$. In this classical case, there have been extensive studies on the estimate of its eigenvalues. Here we mention the work done in [15, 16, 26, 32, 43, 48] as well as the references therein.

If the vector fields in $X$ satisfy the condition (H) on $\overline{\Omega}$ with Hörmander index $Q\geq 2$, Métivier [37] initiated the study on the asymptotic behavior of the eigenvalues under an extra assumption on $X$:

For each $x\in\overline{\Omega}$, let $V_{j}(x)~{}(1\leq j\leq Q)$ be the subspaces of the tangent space at $x$ spanned by all commutators of $X_{1},\ldots,X_{m}$ with length at most $j$. Métivier made the following assumption:

(M): For each $x\in\overline{\Omega}$, $\dim V_{j}(x)$ is a constant (denoted by $\nu_{j}$) in a neighborhood of $x$.

Under the above additional hypothesis (M), Métivier in [37] proved the following asymptotic expression for the sub-elliptic Dirichlet eigenvalue $\lambda_{k}$,

[TABLE]

where $\gamma(x)$ is a positive continuous function on $\Omega$. The index $\nu$ is defined as

[TABLE]

which is called the Métivier index of $\Omega$ (here $\nu$ is also called the Hausdorff dimension of $\Omega$ related to the sub-elliptic metric induced by the vector fields $X$).

The asymptotic formula (1.2) fails to hold for general Hörmander vector fields not satisfying the Métivier condition. To our best knowledge, there is little information in literature about the explicit asymptotic behavior of Dirichlet eigenvalues for general sub-elliptic operators which only satisfy Hörmander’s condition (H). Recently, in the case of $X_{j}^{*}=-X_{j}$, Chen and Luo in [14] estimated the lower bound of $\lambda_{k}$ for the self-adjoint sum of square operator $L=-\sum_{j=1}^{m}X_{j}^{2}$. They proved that

[TABLE]

where $C_{0}$ is a positive constant related to $X$ and $\Omega$. Consequently, (1.4) implies $\lambda_{k}\geq C_{0}\cdot k^{\frac{2}{nQ}}$.

From $\eqref{1-3}$, we can deduce that $n+Q-1\leq\nu\leq nQ$, and actually $\nu=nQ$ if and only if $Q=1$. It can be seen that if $X$ satisfy the condition (M) with Hörmander index $Q>1$, the growth order for $\lambda_{k}$ in (1.4) is $\frac{2}{Qn}$, which is smaller than the one in Métivier’s asymptotic formula (1.2). This shows that Chen and Luo’s lower bound estimate (1.4) is not optimal under the condition (M).

There are many results under the Métivier’s condition (M), such as sub-elliptic estimates and function spaces on nilpotent Lie groups, Sobolev inequalities, Harnack inequality and heat kernel estimates on nilpotent Lie groups (cf. [19, 46]). Parallel to the classical Laplacian $\triangle$ in $\mathbb{R}^{n}$, the Kohn Laplacian operator $\triangle_{\mathbb{H}}$ induced by left invariant vector fields on Heisenberg group $(\mathbb{H}_{n},\circ)$ is a sub-elliptic operator which plays an important role in physics. In 1994, Hansson and Laptev [21] gave a precise lower bounds of Dirichlet eigenvalues $\lambda_{k}$ for the Kohn Laplacian operator $\triangle_{\mathbb{H}}$. The Métivier’s condition posses a strong restriction on the vector fields $X$ satisfying Hörmander’s condition, under which the Lie algebra generated by the vector fields $X_{1},X_{2},\ldots,X_{m}$ takes a constant structure and the vector fields can be well approximated by some homogeneous left invariant vector fields defined on the corresponding Carnot group (cf. [44]). In this paper, we will deal with general self-adjoint Hörmander operators $-\triangle_{X}=\sum_{i=1}^{m}X_{i}^{*}X_{i}$ without the restriction of Métivier’s condition (M). A main purpose is to establish a sharp lower bound of the Dirichlet eigenvalue $\lambda_{k}$ for the sub-elliptic operator $-\triangle_{X}$. Furthermore, we construct an asymptotic formula for $\lambda_{k}$ which is a generalization of Métivier’s result (1.2). In fact, Métivier’s condition (M) is just a sufficient condition for this generalized asymptotic formula. Our discussion below demonstrates that the Métivier’s condition (M) can be relaxed to a weak condition which is now the necessary and sufficient condition for the asymptotic formula of $\lambda_{k}$ being satisfied. Also, under this weak condition, the asymptotic formula shows that our lower bound for $\lambda_{k}$ is optimal in terms of the order on $k$.

In this paper, the general Hörmander vector fields $X$ need not necessary to satisfy the Métivier’s condition (M). Therefore, we need to introduce the following generalized Métivier’s index which is also called the non-isotropic dimension of $\Omega$ related to $X$ (cf. [14, 39, 50]). With the same notations as before, we denote here $\nu_{j}(x)=\dim V_{j}(x)$ and then $\nu(x)$, the pointwise homogeneous dimension at $x$, is given by

[TABLE]

Then we define

[TABLE]

as the generalized Métivier index of $\Omega$. Observe that $n+Q-1\leq\tilde{\nu}<nQ$ for $Q>1$, and $\tilde{\nu}=\nu$ if the Métivier’s condition (M) is satisfied.

In [14], Chen and Luo considered the Grushin vector fields $X=(\partial_{x_{1}},\cdots,\partial_{x_{n-1}},x_{1}^{l}\partial_{x_{n}})$ defined in $\mathbb{R}^{n}$ ($n\geq 2$, $l\in\mathbb{N}^{+}$). The domain $\Omega$ is assumed to be a bounded connected open subset with smooth non-characteristic boundary for $X$ and $\Omega\cap\{x_{1}=0\}\neq\varnothing$. In this case, the Métivier’s condition (M) is not satisfied. However, the vector fields $X$ are finitely degenerate with $Q=l+1\geq 2$ and the generalized Métivier index $\tilde{\nu}=n+Q-1=n+l$. Then the Chen and Luo’s results in [14] gave a sharp lower bound estimates for Dirichlet eigenvalues of $-\triangle_{X}$, i.e. $\lambda_{k}\geq c_{1}k^{\frac{2}{\tilde{\nu}}}$. In [13], the authors further extended this result to more general Grushin type operators.

We now return to our general consideration. Our first goal is to show that the above sharp lower bound is also hold for general sub-elliptic operator $\triangle_{X}$. The key ingredient of our argument is to establish the following uniform upper bound for the Dirichlet heat kernel of sub-elliptic operator $\triangle_{X}$:

Theorem 1.1**.**

Let $X=(X_{1},X_{2},\cdots,X_{m})$ be $C^{\infty}$ real vector fields defined on a connected open domain $W\subset\mathbb{R}^{n}$, which satisfy the condition (H) in $W$. Assume that $\Omega\subset\subset W$ is a bounded connected open subset, and $\partial\Omega$ is smooth and non-characteristic for $X$. If the Hörmander index $Q\geq 2$, then the self-adjoint sub-elliptic operator $\triangle_{X}=-\sum_{i=1}^{m}X_{i}^{*}X_{i}$ has a positive smooth Dirichlet heat kernel $h_{D}(x,y,t)\in C^{\infty}(\Omega\times\Omega\times(0,+\infty))$, which satisfies the following uniform upper bound estimate

[TABLE]

where $\tilde{\nu}$ is the generalized Métivier index of $X$ on $\Omega$, and $C$ is a positive constant depending on $X$ and $\Omega$.

From Theorem 1.1, we can deduce the following sharp lower bound estimate of $\lambda_{k}$ for the sub-elliptic Dirichlet problem (1.1).

Theorem 1.2**.**

Suppose that $X=(X_{1},X_{2},\cdots,X_{m})$ satisfy the same conditions of Theorem 1.1. Then for any $k\geq 1$, we have

[TABLE]

where $\tilde{\nu}$ is the generalized Métivier index of $X$ on $\Omega$, $C_{1}=(C\mbox{e}|\Omega|)^{-\frac{2}{\tilde{\nu}}}$ is a positive constant depending on the volume of $\Omega$ and $\tilde{\nu}$, and $C$ is the same constant as in (1.7).

Furthermore, we obtain the following asymptotic formula for the sub-elliptic Dirichlet eigenvalues $\lambda_{k}$.

Theorem 1.3**.**

Suppose that $X=(X_{1},X_{2},\cdots,X_{m})$ satisfy the same conditions as Theorem 1.1. Then there exists a non-negative measurable function $\gamma_{0}$ on $\overline{\Omega}$ with $\gamma_{0}(x)>0$ for all $x\in\Omega$ such that

[TABLE]

Here $H:=\{x\in\Omega~{}|~{}\nu(x)=\tilde{\nu}\}$ is a subset of $\Omega$, and $N(\lambda):=\#\{k~{}|~{}0<\lambda_{k}\leq\lambda\}$. Moreover, we can deduce that

If $|H|>0$, we have

[TABLE]

• 2.

If $|H|=0$, then we have

[TABLE]

The results of Theorem 1.3 have the following obvious corollary.

Corollary 1.1**.**

For the Dirichlet eigenvalues $\lambda_{k}$ of sub-elliptic operator $-\triangle_{X}$ on $\Omega$. Also $\lambda_{k}\approx k^{\frac{2}{\tilde{\nu}}}$ holds as $k\to+\infty$ if and only if $|H|>0$.

Remark 1.1**.**

We mention that from the Theorem 1.3 if $H=\{x\in\Omega~{}|~{}\nu(x)=\tilde{\nu}\}$ has a positive measure, the lower bound (1.8) for $\lambda_{k}$ in Theorem 1.2 is optimal in terms of the order on $k$. In particular, when Métivier’s condition (M) is satisfied, we know that $H=\Omega$ and the condition $|H|>0$ is certainly satisfied. In this case, our asymptotic formula (1.10) coincides with Métivier’s asymptotic estimate (1.2). If $H$ has zero measure, then the result of Theorem 1.3 implies that our lower bound estimate (1.8) for the eigenvalue $\lambda_{k}$ is not optimal, for $\lambda_{k}^{-1}=o(k^{-\frac{2}{\tilde{\nu}}})$ as $k\to+\infty$.

Remark 1.2**.**

The result of Corollary 1.1 has the following geometric meaning: Under the condition $|H|>0$, the non-isotropic dimension $\tilde{\nu}$ of $\Omega$ related to $X$ will be a spectral invariant.

For upper bounds of Dirichlet eigenvalues $\lambda_{k}$ for sub-elliptic operator $-\triangle_{X}$, we have the following result.

Theorem 1.4**.**

Assume that the real smooth vector fields $X=(X_{1},X_{2},\cdots,X_{m})$ satisfy the same conditions as in Theorem 1.1. Then for any $k\geq 1$ and the $k^{th}$ Dirichlet eigenvalue $\lambda_{k}$ for the sub-elliptic operator $-\triangle_{X}$, we have

[TABLE]

where $\tilde{C}>0$ is a constant depending on $X$ and $\Omega$.

It is well-known that, in the non-degenerate case, the eigenvalues $\lambda_{k}$ of Dirichlet Laplacian have asymptotic behavior $\lambda_{k}\approx k^{\frac{2}{n}}$ as $k\to+\infty$. Thus, the result in (1.12) means that the upper bounds of Dirichlet eigenvalues $\lambda_{k}$ of $-\triangle_{X}$ have the same order in $k$ with that in the non-degenerate case. If the Hörmander index $Q>1$, we have $\tilde{\nu}>n$. Then from Corollary 1.1 above, we know that the upper bounds (1.12) is not optimal in the case of $|H|>0$. However, the following result demonstrates that the result of the upper bounds (1.12) cannot be improved in general in case $|H|=0$. To be more detailed, we introduce the following condition:

(A): We say that the vector fields $X=(X_{1},X_{2},\cdots,X_{m})$ satisfy assumption (A) on $\Omega$ if

[TABLE]

Here the sum is over all $n$-combinations $(Y_{i_{1}},Y_{i_{2}},\cdots,Y_{i_{n}})$ of the set $\{X_{1},X_{2},\cdots,X_{m}\}$.

Actually, we can deduce $|H|=0$ from the condition (A). In fact, for each $x\in H$, since $Q\geq 2$, we have $\nu_{1}(x)=\dim V_{1}(x)<n$. This is because that if $\nu_{1}(x)=n$, then $V_{1}(x)=V_{2}(x)=\cdots=V_{Q}(x)=T_{x}(W)$, which implies that $\nu(x)=\sum_{j=1}^{Q}j(\nu_{j}(x)-\nu_{j-1}(x))=n$, but $\tilde{\nu}\geq n+Q-1\geq n+1>\nu(x)$, that means $x\notin H$. Thus we introduce the set $E$ by $E:=\{x\in\Omega|\sum|\det(Y_{i_{1}},Y_{i_{2}},\cdots,Y_{i_{n}})(x)|=0\}$. Then for any $x\in H$, the fact $\nu_{1}(x)=\dim V_{1}(x)<n$ implies that $\sum|\det(Y_{i_{1}},Y_{i_{2}},\cdots,Y_{i_{n}})(x)|=0$, where the sum is taken over all $n$-combinations $(Y_{i_{1}},Y_{i_{2}},\cdots,Y_{i_{n}})$ of the set $\{X_{1},X_{2},\cdots,X_{m}\}$. Hence we have $H\subset E$.

On the other hand, if we write $g(x)=\sum|\det(Y_{i_{1}},Y_{i_{2}},\cdots,Y_{i_{n}})(x)|\geq 0$, where the sum is taken over all $n$-combinations $(Y_{i_{1}},Y_{i_{2}},\cdots,Y_{i_{n}})$ of the set $\{X_{1},X_{2},\cdots,X_{m}\}$. Thus if we let $A=\int_{\Omega}\frac{dx}{\sum|\det(Y_{i_{1}},Y_{i_{2}},\cdots,Y_{i_{n}})(x)|}<+\infty$, then (1.13) implies that the set $E_{n}=\{x\in\Omega|\frac{1}{g(x)}\geq n\}$ satisfies $|E_{n}|\leq\frac{A}{n}$ for each $n\in\mathbb{N}^{+}$. Observe that $E_{n}=\{x\in\Omega|\frac{1}{g(x)}\geq n\}=\{x\in\Omega|0\leq g(x)\leq\frac{1}{n}\}$ and $E_{n+1}\subset E_{n}$. We then have $E=\{x\in\Omega|g(x)=0\}=\cap_{n=1}^{\infty}E_{n}$. Therefore, $|E|=\lim_{n\to\infty}|E_{n}|=0$. Since $H\subset E$, we obtain $|H|=0$.

Our next result is stated as follows.

Theorem 1.5**.**

If the real smooth vector fields $X=(X_{1},X_{2},\cdots,X_{m})$ satisfy the same conditions of Theorem 1.1 and assumption $(A)$ on $\Omega$, then we have

[TABLE]

Here the constant $C>0$ is independent of $k$, and $\lambda_{i}$ is the $i^{th}$ Dirichlet eigenvalue of problem (1.1).

Remark 1.3**.**

The conclusion of Theorem 1.5 implies that, under condition (A), the Dirichlet eigenvalues $\lambda_{k}$ for a degenerate elliptic operator $-\triangle_{X}$ will have the same asymptotic behavior with the non-degenerate Laplacian case: $\lambda_{k}\approx k^{\frac{2}{n}}$ as $k\to+\infty$. Also in this case, the upper bound (1.12) for $\lambda_{k}$ is optimal in terms of the growth order in $k$.

Remark 1.4**.**

One important study where the system $X$ appears in application is when one studies the CR vector fields of CR manifolds. For simplicity, we let $M$ be a smooth real hypersurface in a complex Euclidean space ${\mathbb{C}}^{n}$ with $n\geq 2$ defined by $\rho=0$. Write $(z_{1},\cdots,z_{n})$ for the coordinates of ${\mathbb{C}}^{n}$. Assume without loss of generality that $\rho_{z_{n}}:=\frac{\partial{\rho}}{\partial{z_{n}}}\not=0$ along $M$. Then $L_{j}=\frac{\partial{}}{\partial{z_{j}}}-\frac{{\rho_{z_{j}}}}{{\rho_{z_{n}}}}\frac{\partial{\ }}{\partial{z_{n}}}$ for $j=1,\cdots,n-1$ form a basis of CR vector fields along $M$. Let $X_{j}=\hbox{Re}(L_{j})$ and $X_{j+n-1}=\hbox{Im}(L_{j})$. Then the system $X=\{X_{1},\cdots,X_{2n-2}\}$ satisfies the Hörmander condition if and only if $M$ is of finite type in the sense of Bloom-Graham that is equivalent to the geometric condition that there is no complex hypersurface contained in $M$ (see the book of Baouendi-Ebenfelt-Rothschild [6] for related references). When $M$ is Levi non-degenerate, then the Hörmander index of $X$ is always $2$ at each point along $M$ and thus the Métivier condition holds. The other situation where the Métivier condition holds is when $M$ has uniform finite non-degeneracy (see the work of Baounendi-Huang-Rothschild [7] for definition and many examples of this type hypersurfaces). For instance, this is the case when $M\subset{\mathbb{C}}^{3}$ is the Freeman cone defined by $z_{1}^{2}+z_{2}^{2}=z_{3}^{2}$. In general, the Métivier condition is rarely satisfied for $X$ with such a geometric background. The generalized Métivier index is associated with the degeneracy of the Levi-form along $M$. It is two if and only if the point is a Levi non-degenerate point along at least one CR direction and is at least three otherwise. The Hörmander sub-elliptic Laplacian associated with $X$ is more or less the Kohn’s sub-Laplacian operator of $M$. There have been much work done to study the spectral theory in the strongly pseudo-convex case (see [8]). Our result in the present paper may shed the light for $M$ being weakly pseudo-convex but of finite type where much less is known. We hope to come back to such an application in a future work.

Remark 1.5**.**

Some other results on eigenvalues of hypoelliptic operators, one can see [34, 35, 36, 45, 38] and references therein.

The plan of the rest paper is as follows. In Section 2, we present some preliminaries including the weighted Sobolev embedding theorem, the weighted Poincaré inequality induced by vector fields $X$, the sub-elliptic estimates, Carnot-Carathéodory metric and the estimate of volume for subunit ball. In Section 3, we establish a supremum norm estimates of the Dirichlet eigenfunction and an explicit lower bound of the Dirichlet eigenvalue. In Section 4, we discuss the existence of the Dirichlet heat kernel for the sub-elliptic operator $\triangle_{X}$ and some basic properties for the fundamental solution of the degenerate heat equation. In Section 5, we study the diagonal asymptotic behavior of the Dirichlet heat kernel for the sub-elliptic operator $\triangle_{X}$. The proofs of Theorem 1.1, Theorem 1.2 and Theorem 1.3 will be given in Section 6, and the proofs of Theorem 1.4 and Theorem 1.5 will be given in Section 7 respectively. Finally, as applications of Theorem 1.2 – Theorem 1.5, we shall present more related examples in Section 8.

2 Preliminaries

2.1 Some estimates on weighted Sobolev spaces.

We start with the following weighted Sobolev embedding theorem.

Proposition 2.1** (Weighted Sobolev Embedding Theorem).**

Let $X=(X_{1},X_{2},\cdots,X_{m})$ be $C^{\infty}$ vector fields defined on a connected open subset $W$ in $\mathbb{R}^{n}$, which satisfy condition (H). Assume that $\Omega\subset\subset W$ is a bounded open subset with smooth boundary $\partial\Omega$ which is non-characteristic for $X$. Denote $\tilde{\nu}$ by the generalized Métivier index of $X$ on $\Omega$. Then for $1\leq p<\tilde{\nu}$, there exists a constant $C=C(\Omega,X)>0$, such that for all $u\in C^{\infty}(\overline{\Omega})$, the inequality

[TABLE]

holds for $q=\frac{\tilde{\nu}p}{\tilde{\nu}-p}$.

Proof.

See Corollary 1 in [50]. ∎

In particular, if $Q\geq 2$, then $\tilde{\nu}\geq n+Q-1\geq 3$. Putting $p=2$ into Proposition 2.1, we can deduce that

[TABLE]

where $u\in H_{X,0}^{1}(\Omega)$.

We also have the following weighted Poincaré inequality for the vector fields $X$.

Proposition 2.2** (Weighted Poincaré Inequality).**

Suppose that $X=(X_{1},X_{2},\cdots,X_{m})$ satisfy the same conditions as in Theorem 1.1. Then the first eigenvalue $\lambda_{1}$ of the Dirichlet problem (1.1) for $-\triangle_{X}$ is positive. Moreover, we have the following weighted Poincaré inequality

[TABLE]

Proof.

We set

[TABLE]

Suppose $\lambda_{1}=0$. Then there exists a sequence $\{\varphi_{j}\}$ in $H_{X,0}^{1}(\Omega)$ such that $\|X\varphi_{j}\|_{L^{2}(\Omega)}\to 0$ with $\|\varphi_{j}\|_{L^{2}(\Omega)}=1$. Since $H_{X,0}^{1}(\Omega)$ is compactly embedded into $L^{2}(\Omega)$ (see [17, 33]), the variational calculus ensures that there exists $\varphi_{0}\in H_{X,0}^{1}(\Omega)$ with $\|\varphi_{0}\|_{L^{2}(\Omega)}=1$ that satisfies $\triangle_{X}\varphi_{0}=0$ and $\|X\varphi_{0}\|_{L^{2}(\Omega)}=0$. The condition (H) implies that $\triangle_{X}$ is hypo-elliptic on $\Omega$. Meanwhile, $\partial\Omega$ is $C^{\infty}$ and non-characteristic for $X$. Thus, we know that $\varphi_{0}\in C^{\infty}(\overline{\Omega})$ and $\varphi_{0}|_{\partial\Omega}=0$ (see [17, 25, 42]). By Bony’s strong maximum principle (see [9, 42]), we can deduce that $\varphi_{0}$ must attain its maximum and minimum values on $\partial\Omega$ unless $\varphi_{0}$ is a constant on $\overline{\Omega}$. Thus we obtain $\varphi_{0}\equiv 0$, which contradicts with $\|\varphi_{0}\|_{L^{2}(\Omega)}=1$. We thus proved that $\lambda_{1}>0$. ∎

Combining (2.2) with (2.3), we obtain the following weighted Sobolev inequality.

Proposition 2.3** (Weighted Sobolev Inequality).**

Suppose that $X=(X_{1},X_{2},\cdots,X_{m})$ satisfy the same conditions as in Theorem 1.1. Then there exists a constant $C=C(\Omega,X)>0$, such that for any $u\in H_{X,0}^{1}(\Omega)$ we have

[TABLE]

Also, we need the following sub-elliptic estimates.

Proposition 2.4** (Sub-elliptic estimates I).**

Assume that $X=(X_{1},X_{2},\cdots,X_{m})$ satisfy the condition (H) on an open domain $W$ in $\mathbb{R}^{n}$. Then, for any open subset $\Omega\subset\subset W$, there exist constants $\epsilon_{0}>0$ and $C>0$ such that

[TABLE]

Proof.

See Theorem 17 in [44]. ∎

Proposition 2.5** (Sub-elliptic estimates II).**

Suppose that $X=(X_{1},X_{2},\cdots,X_{m})$ satisfy the condition (H) on an open domain $W$ in $\mathbb{R}^{n}$. Let $\Omega\subset\subset W$ be an open subset and $\phi\prec\phi_{1}$ be nested cut-off functions with support in $\Omega$ (i.e. $\phi,~{}\phi_{1}\in C_{0}^{\infty}(\Omega)$, and $\phi_{1}\equiv 1$ on the support of $\phi$ ). Then there exists $\epsilon>0$ so that for every $s\geq 0$, there is a constant $C>0$ such that

[TABLE]

holds for any $u\in L^{2}(\Omega)\cap C^{\infty}(\Omega)$.

Proof.

See Theorem 17.0.1 in [40], Theorem 18 in [44] and also refer to [24]. ∎

From the Sobolev imbedding theorem we know that for $s>\frac{n}{2}$, there exists a constant $C>0$ such that

[TABLE]

Thus, combining (2.7) with Proposition 2.5, we have following corollary.

Corollary 2.1**.**

Let $N\in\mathbb{N}^{+}$ with $N>\frac{n}{2\epsilon}$ (where $\epsilon$ was given in Proposition 2.5) and $\xi(x)\in C_{0}^{\infty}(\Omega)$. If $u\in L^{2}(\Omega)\cap C^{\infty}(\Omega)$ and $(\triangle_{X})^{k}u\in L^{2}(\Omega)$ for $1\leq k\leq N$, then we have

[TABLE]

Proof.

See Corollary 17.0.2 in [40]. ∎

2.2 Carnot-Carathéodory metric and volume of subunit ball.

We briefly introduce some geometric properties of the metric induced by vector fields $X$ in this part. Readers can refer to [18],[41] and [39] in details.

Let $X=(X_{1},X_{2},\cdots,X_{m})$ satisfy the condition (H) on a connected open set $V\subset\mathbb{R}^{n}$ with Hörmander’s index $r_{0}$. Then the subunit metric (also known as Carnot-Carathéodory metric, or control distance) can be defined as follows.

For $x,y\in V$ and $\delta>0$, let $C(x,y,\delta)$ denote the collection of absolutely continuous mapping $\varphi:[0,1]\mapsto V$, which satisfying $\varphi(0)=x,\varphi(1)=y$ and

[TABLE]

with $\sum_{k=1}^{m}|a_{k}(t)|^{2}\leq\delta^{2}$ for a.e $t\in[0,1]$. From the Chow-Rashevskii theorem (See [10], Theorem 57) we know that there exist a $\delta>0$ such that $C(x,y,\delta)\neq\varnothing$. Then we can define the subunit metric $d_{X}(x,y)$ as follows

[TABLE]

Now, we denote

[TABLE]

as the subunit ball induced by the subunit metric $d_{X}(x,y)$. For the volume of the subunit ball, a well-known result by Fefferman and Phong [18] states that for any compact set $K\subset V$, there are constants $c=c(K)>0,\delta_{0}=\delta_{0}(K)>0$ and $\epsilon_{0}>0$ such that for any $x\in K$ and $0<r<\delta_{0}$ we have

[TABLE]

where $B(x,r)$ is the ball in the classical Euclidean metric. Moreover, we can precisely estimate the volume of the subunit ball by Proposition 2.6 below.

Since $X_{1},\ldots,X_{m}$ together with their commutators of length at most $r_{0}$ span $T_{x}(V)$ at each point $x$ of $V$, we can write the commutators of higher order by means of the following standard notation.

Let $I=(j_{1},\ldots,j_{k})$ ($1\leq j_{i}\leq m$) is a multi-index with length $|I|=k$,

[TABLE]

The set $X^{(k)}$ is defined as commutators of length $k$:

[TABLE]

[TABLE]

[TABLE]

Let $Y_{1},\cdots,Y_{q}$ be some enumeration of the components of $X^{(1)},\ldots,X^{(r_{0})}$. If $Y_{i}$ is an element of $X^{(k)}$, we say $Y_{i}$ has formal degree $d(Y_{i})=k$. By notation in [41], for each $n$-tuple of integers $I=(i_{1},\ldots,i_{n}),1\leq i_{j}\leq q$, we set

[TABLE]

(If $Y_{i_{j}}=\sum_{k=1}^{n}a_{jk}(x)\partial_{x_{k}}$, then $\det(Y_{i_{1}},\cdots,Y_{i_{n}})(x)=\det(a_{jk}(x))$). We also set

[TABLE]

then we define the $\Lambda(x,r)$ as

[TABLE]

where the sum is taken over all $n$-tuples. Now we state the following proposition obtained by Nagel, Stein and Wainger.

Proposition 2.6** (Ball-Box theorem).**

For any compact set $K\subset V$, there exists $\delta_{0}=\delta_{0}(K)>0$, and $C_{1},C_{2}>0$ such that for all $x\in K$ and $r\leq\delta_{0}$ we have

[TABLE]

where $|B_{d_{X}}(x,r)|$ is the Lebesgue measure of $B_{d_{X}}(x,r)$.

Proof.

See [39] and [41]. ∎

According to (2.11) and Proposition 2.6, we can deduce that the pointwise homogeneous dimension of $x$ has the following property.

[TABLE]

Then from the (2.11),(2.12) and (2.13), we know that $|B_{d_{X}}(x,r)|$ behaves like $r^{\nu(x)}$ as $r\to 0^{+}$.

3 Explicit estimates of Dirichlet eigenfunctions and Dirichlet eigenvalues

3.1 Supremum norm estimates of Dirichlet eigenfunctions

The task in this part is to estimate the supremum norm of Dirichlet eigenfunctions for sub-elliptic operator $-\triangle_{X}=\sum_{i=1}^{m}X_{i}^{*}X_{i}$.

For each $i\in\mathbb{N}^{+}$, $\phi_{i}\in H_{X,0}^{1}(\Omega)$ denotes as the $i^{th}$ Dirichlet eigenfunction corresponding with the $i^{th}$ Dirichlet eigenvalue $\lambda_{i}$, we have $(\triangle_{X}+\lambda_{i})\phi_{i}=0$. According to the regularity results of Derridj in [17], we know that $\phi_{i}\in C^{\infty}(\overline{\Omega})$ and $\phi_{i}|_{\partial\Omega}=0$. Moreover, the sequence of eigenfunctions $\{\phi_{i}\}_{i=1}^{\infty}$ constitutes an orthogonal basis in $H_{X,0}^{1}(\Omega)$ with $\|\phi_{i}\|_{L^{2}(\Omega)}=1$, which is also a standard orthogonal basis in $L^{2}(\Omega)$. Furthermore, we have the following estimates of $L^{\infty}$-norm for the Dirichlet eigenfunction $\phi_{i}$.

Proposition 3.1**.**

Suppose that $X=(X_{1},X_{2},\cdots,X_{m})$ satisfy the conditions of Theorem 1.1. We have

[TABLE]

where $C_{1}$ is a positive constant depending on $X$ and $\Omega$, $\tilde{\nu}$ is the generalized Métivier index on $\Omega$, $\|\cdot\|_{\infty}$ denotes the $L^{\infty}$-norm on $\Omega$.

Proof.

Since $-\triangle_{X}\phi_{i}=\lambda_{i}\phi_{i}$, then

[TABLE]

For any constant $s\geq 2$, we take $u=|\phi_{i}|^{s-1}\cdot\text{sgn}(\phi_{i})$. Since

[TABLE]

we can deduce that $u\in H_{X,0}^{1}(\Omega)$. Therefore (3.2) implies that

[TABLE]

For each $f\in H_{X,0}^{1}(\Omega)$, we know that $X|f|=\text{sgn}(f)Xf$ (cf. [20] Lemma 3.5). Moreover, (3.3) gives

[TABLE]

On the other hand, for any non-negative function $v\in H_{X,0}^{1}(\Omega)\cap L^{\infty}(\Omega)$ and any constant $s\geq 2$, integrating by parts and applying the weighted Sobolev inequality (2.4) we have

[TABLE]

where $C$ is the Sobolev constant in (2.4). Thus if $v=|\phi_{i}|$, then $v\in H_{X,0}^{1}(\Omega)\cap L^{\infty}(\Omega)$. Hence (3.4) and (3.1) assert that

[TABLE]

which can be rewritten as

[TABLE]

for all $s\geq 2$, with $\beta=\frac{\tilde{\nu}}{\tilde{\nu}-2}$. Here $\|\phi_{i}\|_{s}$ is the $L^{s}$-norm of $\phi_{i}$. Setting $s=2\beta^{j}\geq 2$, respectively for $j=0,1,2,\cdots$, then we have

[TABLE]

Iterating this estimate and using $\|\phi_{i}\|_{2}=1$, we conclude that

[TABLE]

Letting $j\to\infty$ and applying the fact that $\lim_{p\to\infty}\|\phi_{i}\|_{p}=\|\phi_{i}\|_{\infty}$, we obtain

[TABLE]

where $C_{1}$ is a positive constant depends on $C$ and $\tilde{\nu}$. ∎

3.2 An explicit lower bound of Dirichlet eigenvalues

The aim in this part is to get an explicit lower bound of the sub-elliptic Dirichlet eigenvalue $\lambda_{k}$. Although the lower bound of $\lambda_{k}$ may not be precise, it is useful in the process of estimating Dirichlet heat kernel of $\triangle_{X}$.

Proposition 3.2**.**

Suppose $X=(X_{1},X_{2},\cdots,X_{m})$ satisfy the conditions of Theorem 1.1. Then we have

[TABLE]

where the positive constant $C$ depends on vector fields $X$ and $\Omega$, and $\epsilon_{0}$ is a positive constant in Proposition 2.4.

Our proof of Proposition 3.2 is inspired by Chen and Luo’s approach in [14] and the work of Li and Yau in [32]. We need several lemmas to prove Proposition 3.2.

Lemma 3.1**.**

Let $\{\phi_{j}\}_{j=1}^{k}$ be the set of orthonormal eigenfunctions corresponding to the Dirichlet eigenvalues $\{\lambda_{j}\}_{j=1}^{k}$. Define

[TABLE]

Then we have

[TABLE]

where $\hat{\Phi}(z,y)$ is the partial Fourier transformation of $\Phi(x,y)$ in the $x$-variable

[TABLE]

Proof.

See Lemma 3.1 in [14]. ∎

Lemma 3.2**.**

Let $f$ be a real-valued function defined on $\mathbb{R}^{n}$ with $0\leq f\leq M_{1}$. If

[TABLE]

with $\epsilon_{0}>0$, then we have the following inequality,

[TABLE]

where $B_{n}$ is the volume of the unit ball in $\mathbb{R}^{n}$.

Proof.

First, we can choose $R$ such that

[TABLE]

where

[TABLE]

Then $(|z|^{2\epsilon_{0}}-R^{2\epsilon_{0}})(f(z)-g(z))\geq 0$. Hence we get

[TABLE]

Note that

[TABLE]

where $\omega_{n-1}$ is the area of the unit sphere in $\mathbb{R}^{n}$. By the definition of $g(z)$, we know

[TABLE]

where $B_{n}$ is the volume of the unit ball in $\mathbb{R}^{n}$.

Since $nB_{n}=\omega_{n-1}$, then (3.7),(3.8) and (3.9) give

[TABLE]

∎

Now, we can prove Proposition 3.2.

Proof of Proposition 3.2 .

For $\Phi(x,y)=\sum_{j=1}^{k}\phi_{j}(x)\phi_{j}(y)$, we know that $\Phi(x,y)\in H_{X,0}^{1}(\Omega)$ with respect to $x$. By Proposition 2.4 we can deduce that

[TABLE]

where $\nabla=(\partial_{x_{1}},\ldots,\partial_{x_{n}})$, $|\nabla|^{\epsilon_{0}}$ is a pseudo-differential operator with the symbol $|\xi|^{\epsilon_{0}}$, $C>0$ is a constant depends on $X$ and $\Omega$, and $\epsilon_{0}$ is a positive constant in Proposition 2.4. Then, by using Placherel’s formula, we have

[TABLE]

Combining (3.10) and (3.11), we get

[TABLE]

On the other hand, we can deduce that

[TABLE]

It follows from estimates (3.11), (3.12) and (3.2) that

[TABLE]

Now we take

[TABLE]

Then, due to Lemma 3.1 and Lemma 3.2, we have

[TABLE]

Consequently,

[TABLE]

Therefore, by $\lambda_{i}\leq\lambda_{i+1}$ we have

[TABLE]

∎

4 Sub-elliptic Dirichlet heat kernel

We construct the sub-elliptic Dirichlet heat kernel of $\triangle_{X}$ in this section. Our approach is similar to that in Li’s work [31] in the classical case. The sub-elliptic Dirichlet heat kernel $h_{D}(x,y,t)$ of $\triangle_{X}$ is the fundamental solution of the degenerate heat operator $\partial_{t}-\triangle_{X}$. That is, for any fixed point $y\in\Omega$, $h_{D}(x,y,t)$ is the solution of

[TABLE]

and satisfies following properties

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since the Dirichlet heat kernel $h_{D}(x,y,t)$ is the fundamental solution of $\frac{\partial}{\partial t}-\triangle_{X}$. Thus for a function $f_{0}(x)\in L^{2}(\Omega)$, the function

[TABLE]

will solve the degenerate heat equation

[TABLE]

and satisfies

[TABLE]

Recall that the sequence of eigenfunctions $\{\phi_{i}\}_{i=1}^{\infty}$ is a standard orthogonal basis in $L^{2}(\Omega)$, that implies that a function $f_{0}\in L^{2}(\Omega)$ can be written in the form

[TABLE]

Formally, the function $f(x,t)$ can be given by

[TABLE]

which satisfies the (4.8) with initial-boundary condition (4.9). Comparing (4.7) and (4.10), we can deduce that the Dirichlet heat kernel of $\triangle_{X}$ on $\Omega$ can be defined as

[TABLE]

In fact, we have the following proposition.

Proposition 4.1**.**

Let $X=(X_{1},\cdots,X_{m})$ with conditions the same as Theorem 1.1. Then the sub-elliptic operator $\triangle_{X}$ has a Dirichlet heat kernel $h_{D}(x,y,t)$ which is well defined on $\overline{\Omega}\times\overline{\Omega}\times(0,+\infty)$ by

[TABLE]

Furthermore, $h_{D}(x,y,t)$ is uniquely determined and satisfies properties (4.1) to (4.6).

Proof.

We begin by establishing the uniform convergence of the series (4.11). By Proposition 3.1 and recall that $\phi_{i}\in C^{\infty}(\overline{\Omega})$, we have, for $t>0$,

[TABLE]

Now, we use the inequality (cf. [12] Chapter VII)

[TABLE]

Putting $z=\frac{1}{2}\lambda_{i}t$, $\alpha=\frac{\tilde{\nu}}{2}$ into (4.13), we get

[TABLE]

Hence, (4.12) and (4.14) imply that

[TABLE]

where $C_{1}>0$ is a constant depending on $C$ and $\tilde{\nu}$. The explicit lower bound of Dirichlet eigenvalue $\lambda_{k}$ which is established in Proposition 3.2 allows us to obtain that

[TABLE]

where $\epsilon_{0}$ and $C$ are positive constants in Proposition 3.2. The estimate (4.16) implies the series (4.11) uniformly convergent on $\overline{\Omega}\times\overline{\Omega}\times[a,\infty)$ for any $a>0$. Thus the sub-elliptic Dirichlet heat kernel $h_{D}(x,y,t)=\sum_{i=1}^{\infty}e^{-\lambda_{i}t}\phi_{i}(x)\phi_{i}(y)$ is well-defined. Moreover, it can be clearly seen that $h_{D}(x,y,t)$ satisfies (4.4).

We denote $Sh_{N}(x,y,t)$ as the sum of the first $N$ terms of the series (4.11), i.e.

[TABLE]

Since

[TABLE]

Similarly, for any fixed $t>0$, we have

[TABLE]

Thus, it gives us that $Sh_{N}(x,y,t)\to h_{D}(x,y,t)$ uniformly as $N\to+\infty$ in $H_{X,0}^{1}(\Omega)$ for $t>0$. Consequently, for any fixed $(y,t)\in\Omega\times(0,+\infty)$, $h_{D}(x,y,t)\in H_{X,0}^{1}(\Omega)$ with respect to $x$.

Furthermore, for a fixed point $y\in\Omega$ and $N\in\mathbb{Z}^{+}$, $u_{y,N}(x,t)=Sh_{N}(x,y,t)$ is a solution of the degenerate heat equation (4.1). The uniform convergence of $u_{y,N}(x,t)$ implies that $h_{D}(x,y,t)$ is a weak solution of (4.1) with respect to $(x,t)$. Analogously, it is easy to verify that $h_{D}(x,y,t)$ is also a weak solution of equation $[\partial_{t}-\frac{1}{2}(\triangle_{X}^{x}+\triangle_{X}^{y})]u(x,y,t)=0$, since for each $N$, $Sh_{N}(x,y,t)$ is a solution of $[\partial_{t}-\frac{1}{2}(\triangle_{X}^{x}+\triangle_{X}^{y})]u(x,y,t)=0$. Then the hypo-ellipticity of $\partial_{t}-\frac{1}{2}(\triangle_{X}^{x}+\triangle_{X}^{y})$ implies that $h_{D}(x,y,t)\in C^{\infty}(\Omega\times\Omega\times(0,+\infty))$. Also, the uniform convergence of $Sh_{N}(x,y,t)$ on $\overline{\Omega}\times\overline{\Omega}\times[a,+\infty)$ for any $a>0$ gives $h_{D}(x,y,t)\in C(\overline{\Omega}\times\overline{\Omega}\times(0,+\infty))$.

Now recall that the sequence of Dirichlet eigenfunctions $\{\phi_{k}\}_{k=1}^{\infty}$ constitutes a standard orthogonal basis in $L^{2}(\Omega)$. Given a function $f_{0}(x)\in L^{2}(\Omega)$, we have

[TABLE]

where $a_{i}=\int_{\Omega}f_{0}(y)\phi_{i}(y)dy$. In terms of Parseval’s identity we know

[TABLE]

Furthermore, for any $t>0$, we have

[TABLE]

Since

[TABLE]

Then by using similar approach as above, we know that $\sum_{i=1}^{\infty}e^{-\lambda_{i}t}a_{i}\phi_{i}(x)$ converges uniformly on $\overline{\Omega}\times[a,+\infty)$ for any $a>0$, which implies $f(x,t)$ is a weak solution of the degenerate heat equation (4.8) and agrees with the Dirichlet boundary condition in (4.9). Moreover, the hypo-ellipticity of $\partial_{t}-\triangle_{X}$ tells us $f(x,t)\in C^{\infty}(\Omega\times(0,+\infty))\cap C(\overline{\Omega}\times(0,+\infty))$.

In order to verify that $f(x,t)$ satisfies the initial condition in (4.9), it suffices to prove that $f(x,t)=\int_{\Omega}h_{D}(x,y,t)f_{0}(y)dy\to f_{0}(x)$ as $t\to 0^{+}$ in $L^{2}(\Omega)$. It derives, in fact, that

[TABLE]

Thus identity (4.18) implies that $\sum_{i=1}^{\infty}a_{i}^{2}(1-e^{-\lambda_{i}t})^{2}$ converges uniformly on $t\in[0,+\infty)$. Therefore, we obtain

[TABLE]

which means that $f(x,t)$ allows the initial condition in (4.9).

If we take $u(x,t)=\int_{\Omega}h_{D}(x,y,t)\varphi(y)dy-\varphi(x)$ for any $\varphi\in C_{0}^{\infty}(\Omega)$, then for any $t>0$, we have $u(x,t)\in L^{2}(\Omega)\cap C^{\infty}(\Omega)$. Moreover, the symmetry of $h_{D}(x,y,t)$ in $x$ and $y$ gives

[TABLE]

Thus, we know that $(\triangle_{X})^{k}u(x,t)\in L^{2}(\Omega)$ for $t>0$, and $k\in\mathbb{N}^{+}$. Meanwhile, by Corollary 2.1, we have for any $\xi(x)\in C_{0}^{\infty}(\Omega)$,

[TABLE]

Hence from (4.7) and (4.9), the estimate (4) shows that for any cut-off function $\xi(x)\in C_{0}^{\infty}(\Omega)$ we have

[TABLE]

Since the cut-off function $\xi(x)$ is arbitrary, then for any given $x\in\Omega$, (4.20) gives that

[TABLE]

This completes the proof of (4.3).

Also, we have for $t>0$ and $s>0$,

[TABLE]

which yields to (4.5).

Finally, we only need to verify (4.6) and the uniqueness of $h_{D}(x,y,t)$.

We firstly show that $h_{D}(x,y,t)\geq 0$. Actually, if there exists $(x_{0},y_{0},t_{0})\in(\Omega\times\Omega\times(0,+\infty))$ in which $h_{D}(x_{0},y_{0},t_{0})<0$, then there exist $0<\delta<t_{0}$ and $\alpha>0$, such that $B(x_{0},\delta)\subset\Omega$, $B(y_{0},\delta)\subset\Omega$ and for each $(x,y,t)\in B(x_{0},\delta)\times B(y_{0},\delta)\times(t_{0}-\delta,t_{0}+\delta)$, we have

[TABLE]

Thus, we can find a function $f_{0}\in C_{0}^{\infty}(B(y_{0},\delta))$ with $0\leq f_{0}\leq 1$, such that

[TABLE]

Then

[TABLE]

In particular, we have

[TABLE]

Given $U_{T}=\Omega\times(0,T]$ for some $T>t_{0}$. From above arguments, we can conclude that $f(x,t)\in C(\overline{U_{T}})\cap C^{\infty}(U_{T})$. Since $f(x,t)\geq 0$ in the parabolic boundary $\partial_{p}U_{T}=(\Omega\times\{0\})\cup(\partial\Omega\times[0,T])$, it implies that $f(x,t)\geq 0$ in $\Omega\times(0,T)$ according to the weak maximum principle for the degenerate parabolic equation (cf. Proposition 2.2 in [29], also see Proposition 3.6 in [11]). This is a contradiction with (4.21). Hence, we obtain $h_{D}(x,y,t)\geq 0$ for $(x,y,t)\in\Omega\times\Omega\times(0,+\infty)$.

Secondly, we assume that $h_{D}(x^{\prime},y^{\prime},t^{\prime})=0$ for some $(x^{\prime},y^{\prime},t^{\prime})\in\Omega\times\Omega\times(0,+\infty)$. Since $u(x,t)=h_{D}(x,y^{\prime},t)\in C^{\infty}(\Omega\times(0,+\infty))$ satisfies $(\partial_{t}-\triangle_{X})u(x,t)=0$, and $u(x,t)\geq 0$, the Bony’s parabolic type strong maximum principle (see [9] Theorem 3.2, also refer [10, 42]) shows that $u(x,t)\equiv 0$ for all $0<t\leq t^{\prime}$ and all $x\in\Omega$. Now take a function $f\in C_{0}^{\infty}(\Omega)$ such that $f(y^{\prime})\neq 0$, then we have $\lim_{t\to 0^{+}}\int_{\Omega}h_{D}(x,y^{\prime},t)f(x)dx=f(y^{\prime})$ and yet it is contradictory since $u(x,t)=h_{D}(x,y^{\prime},t)\equiv 0$ for all $0<t\leq t^{\prime}$ and all $x\in\Omega$. Hence we eventually obtain $h_{D}(x,y,t)>0$ for all $(x,y,t)\in\Omega\times\Omega\times(0,+\infty)$.

Let $\Omega=\bigcup_{i=1}^{\infty}K_{i}$ with $K_{i}\subset K_{i+1}^{\circ}$. Here $K_{i}$ is compact set and $K_{i}^{\circ}$ the interior of $K_{i}$. Then we define a sequence of functions $f_{i}$ as

[TABLE]

It is easy to verify that $\lim_{i\to\infty}f_{i}(x)=\chi_{\Omega}(x)$, and $0\leq\chi_{K_{i}}(x)\leq f_{i}(x)\leq f_{i+1}(x)\leq\chi_{\Omega}(x)$. Using the weak maximum principle again, we obtain

[TABLE]

Then by Lebesgue’s monotone convergence theorem, we have

[TABLE]

Hence we complete the proof of (4.6).

Besides, if $\overline{f}(x,t)$ is another solution of (4.8) with the same initial condition $f_{0}$, then the weak maximum principle indicates that the solution $f(x,t)-\overline{f}(x,t)$ of (4.8) must be identically equal to [math] since it vanishes on $(\Omega\times\{0\})\cup(\partial\Omega\times[0,+\infty))$. This leads to the uniqueness of $h_{D}(x,y,t)$.

The arguments of all above complete the proof of Proposition 4.1. ∎

5 Diagonal asymptotic of sub-elliptic Dirichlet heat kernel

In this section, we study the diagonal asymptotic behavior of sub-elliptic Dirichlet heat kernel of $\triangle_{X}$. First, by using the following proposition we can extend vector fields $X$ into whole space $\mathbb{R}^{n}$.

Proposition 5.1**.**

Let $X=(X_{1},X_{2},\cdots,X_{m})$ be a system of $C^{\infty}$ vector fields defined in a bounded connected open set $W_{0}\subset\mathbb{R}^{n}$ and satisfying the condition (H) in $W_{0}$. Then, for any connected open sets $\Omega_{1}\subset\subset\Omega_{2}\subset\subset W_{0}$, there exists a new system of $C_{b}^{\infty}$ vector fields $X^{\prime}=(Z_{1},Z_{2},\cdots,Z_{q})~{}(q=m+n)$, such that the vector fields $X^{\prime}$ are defined in the whole space $\mathbb{R}^{n}$ and satisfy the Hörmander’s condition (H) in $\mathbb{R}^{n}$ (actually the vector fields $X^{\prime}$ satisfy the uniform version of Hörmander’s condition in $\mathbb{R}^{n}$, a detail proof will be given in Section 9, Proposition 9.1 below). Moreover

[TABLE]

Furthermore, denoting by $d_{X^{\prime}},d$, respectively, the subunit metric induced by $X^{\prime}$ in $\mathbb{R}^{n}$ and $X$ in $W_{0}$, then for any connected open set $\Omega\subset\subset\Omega_{1}$, $d_{X^{\prime}}$ is equivalent to $d$ in $\Omega$, and $d_{X^{\prime}}$ is equivalent to the Euclidean distance in $\mathbb{R}^{n}\setminus W_{0}$.

Proof.

See Theorem 2.9 in [11]. ∎

Since $\overline{\Omega}$ is a compact subset of $W$, we can always find a bounded connected open set $W_{0}$ which has compact closure $\overline{W_{0}}$ such that $\overline{\Omega}\subset W_{0}\subset\overline{W_{0}}\subset W$. Also, there exists two connected open sets $\Omega_{1},\Omega_{2}$ such that $\Omega\subset\subset\Omega_{1}\subset\subset\Omega_{2}\subset\subset W_{0}$. Therefore, from Proposition 5.1, we get a system of $C_{b}^{\infty}$ vector fields $X^{\prime}=(Z_{1},Z_{2},\cdots,Z_{q})$ which is an extension of vector fields $X$ in $\mathbb{R}^{n}$ and satisfy the uniform Hörmander’s condition. Let $\triangle_{X^{\prime}}=-\sum_{j=1}^{q}Z_{j}^{*}Z_{j}$ be the sub-elliptic operator given by the vector fields $X^{\prime}$, then $\triangle_{X^{\prime}}=\triangle_{X}$ on $\Omega_{1}$ which is a neighborhood of $\overline{\Omega}$. For the sub-elliptic operator $\triangle_{X^{\prime}}$ in $\mathbb{R}^{n}$, by the results in [11, 28], we know it has a global heat kernel $h(x,y,t)$ defined on $\mathbb{R}^{n}\times\mathbb{R}^{n}\times(0,+\infty)$ such that

[TABLE]

and also satisfies the following properties

[TABLE]

[TABLE]

[TABLE]

Meanwhile, the hypoellipticity of $\partial_{t}-\triangle_{X^{\prime}}$ implies that $h(x,y,t)\in C^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n}\times(0,+\infty))$.

For the global heat kernel $h(x,y,t)$, we recall the asymptotic result constructed by Takanobu [47]. Other similar results were also obtained by Ben Arous and Léandre [3, 4, 5].

Proposition 5.2**.**

For the global heat kernel $h(x,y,t)$ of the sub-elliptic operator $\triangle_{X^{\prime}}=-\sum_{j=1}^{q}Z_{j}^{*}Z_{j}$, there exists a sequence of real measurable functions $\{c_{i}(x)\}_{i=1}^{\infty}$ defined in $\mathbb{R}^{n}$, such that $h(x,x,t)$ has following asymptotic formula for $t\to 0^{+}$

[TABLE]

where $c_{0}(x)>0$ for all $x\in\mathbb{R}^{n}$, $\nu(x)$ is the pointwise homogeneous dimension at $x$.

Proof.

See Theorem 6.8 in [47]. ∎

We also need the following Gaussian bounds of global heat kernel $h(x,y,t)$ which was proved by Kusuoka and Stroock in [27, 28] and was also generalized by Brandolini, Bramanti and Lanconelli et al [11] to more general sub-elliptic operators. The similar results over compact manifolds was constructed by Jerison and Sánchez-Calle in [23].

Proposition 5.3**.**

For the global heat kernel $h(x,y,t)$ of the sub-elliptic operator $\triangle_{X^{\prime}}=-\sum_{j=1}^{q}Z_{j}^{*}Z_{j}$, there exist positive constants $A_{1},A_{2},B_{1},B_{2}$ such that for all $(x,y,t)\in\mathbb{R}^{n}\times\mathbb{R}^{n}\times(0,1]$, we have

[TABLE]

where $B_{d_{X^{\prime}}}(x,r)=\{y\in\mathbb{R}^{n}|d_{X^{\prime}}(x,y)<r\}$.

Proof.

See [27], [28] and [11]. ∎

Then, we have the following diagonal asymptotic result of sub-elliptic Dirichlet heat kernel.

Proposition 5.4**.**

Let $h_{D}(x,y,t)$ be the sub-elliptic Dirichlet heat kernel of $\triangle_{X}$ on $\Omega$. Then there exists a non-negative measurable function $\gamma_{0}$ on $\overline{\Omega}$ which satisfies $\gamma_{0}(x)>0$ for all $x\in\Omega$, such that

[TABLE]

Proof.

Let $X^{\prime}$ be a global extension of $X$ in $\mathbb{R}^{n}$ and $h(x,y,t)$ be the corresponding global heat kernel in $\mathbb{R}^{n}$. Given

[TABLE]

it follows from (4.3) and (5.1) that for any $\varphi\in C_{0}^{\infty}(\Omega)$, we have

[TABLE]

Similar to the arguments in the proof of (4.6), we have that

[TABLE]

Also, it is easy to show that for any fixed $y\in\Omega$, $u_{y}(x,t)=E(x,y,t)$ is locally integrable on $\Omega\times\mathbb{R}$ and $u_{y}(x,t)$ satisfies

[TABLE]

in the sense of distribution. Then the hypoellipticity of $\partial_{t}-\triangle_{X}$ implies that for any fixed $y\in\Omega$, $u_{y}(x,t)=E(x,y,t)\in C^{\infty}(\Omega\times\mathbb{R})$. Moreover, for any $x\in\overline{\Omega}$ we have

[TABLE]

Now, for any fixed $y\in\Omega$, we know that $E(x,y,t)\in C^{\infty}(\Omega\times(0,+\infty))\cap C(\overline{\Omega}\times[0,+\infty))$. Then, by using the weak maximum principle, Proposition 5.3 and (2.9), we have for sufficient small $t\ll 1$ and all $x\in\Omega$

[TABLE]

where $C$ is a positive constant depends on $A_{2}$ and $\overline{\Omega}$, $\epsilon_{0}>0$ is the constant in Fefferman and Phong’s estimate (2.9), $\text{dist}_{d_{X^{\prime}}}(y,\partial\Omega):=\inf\{d_{X^{\prime}}(x,y)|x\in\partial\Omega\}$. Now, we define $C_{2}:=C_{2}(y)=B_{2}\text{dist}_{d_{X^{\prime}}}^{2}(y,\partial\Omega)>0$. Observe the function $g(s)=s^{-\frac{n}{2\epsilon_{0}}}e^{-\frac{C_{2}}{s}}$ satisfying $\lim_{s\to 0^{+}}g(s)=\lim_{s\to+\infty}g(s)=0$, and $g^{\prime}(s)=e^{-\frac{C_{2}}{s}}s^{-\frac{n}{2\epsilon_{0}}-2}\left(C_{2}-\frac{n}{2\epsilon_{0}}s\right)$. Therefore, $g(s)$ can only attain its maximum value at $s=\frac{2\epsilon_{0}C_{2}}{n}>0$. Then, for any fixed $y\in\Omega$ and sufficient small $0<t\ll 1$, we have

[TABLE]

In particular, taking $x=y\in\Omega$, we have

[TABLE]

Consequently

[TABLE]

Thus, by Proposition 5.2, there exists a measurable function $c_{0}(x)$ in $\mathbb{R}^{n}$ such that

[TABLE]

We then show that the value of function $c_{0}(x)$ at each point $x\in\Omega$ is independent of the extension of vector fields $X^{\prime}$. If $\widetilde{X}$ is another global extension of $X$ in $\mathbb{R}^{n}$, by the same approach, we also have

[TABLE]

Here $\widetilde{h}(x,y,t)$ is the global heat kernel corresponding with vector fields $\widetilde{X}$, $\widetilde{\epsilon_{0}}$ is a positive constant depends on $\widetilde{X}$. $\widetilde{C}_{2}(x)=\widetilde{B}_{2}\text{dist}_{d_{\widetilde{X}}}^{2}(x,\partial\Omega)$ is a positive constant depends on $x$ and the subunit metric induced by $\widetilde{X}$. It follows from (5.9) and (5.10) that for sufficient small $t$ and all $x\in\Omega$, we have

[TABLE]

Thus

[TABLE]

That implies the value of function $c_{0}(x)$ at each point $x\in\Omega$ is independent of the way of extension.

Finally, we take

[TABLE]

Then we obtain

[TABLE]

∎

6 Proofs of Theorem 1.1, Theorem 1.2 and Theorem 1.3

6.1 Proof of Theorem 1.1.

Proof.

By the semi-group property of $h_{D}(x,y,t)$ in (4.5), we have

[TABLE]

Since $h_{D}(x,y,t)=h_{D}(y,x,t)$, then we obtain

[TABLE]

Moreover

[TABLE]

The last inequality applies the weighted Sobolev inequality (Proposition 2.3) , which is valid since for any fixed $(x,t)\in\Omega\times(0,+\infty)$, $h_{D}(x,y,t)\in H_{X,0}^{1}(\Omega)$ with respect to $y$.

Now, it follows from (4.6) that

[TABLE]

Then the Hölder’s inequality yields

[TABLE]

Hence (6.1),(6.2) and (6.3) give

[TABLE]

For any fixed $x\in\Omega$, take $f(t):=h_{D}(x,x,2t)$ with $t>0$. The positivity of $h_{D}(x,y,t)$ implies that $f(t)>0$. Then it follows from (5.7) and (6.4) that

[TABLE]

Let

[TABLE]

Then

[TABLE]

Now integrating $g^{\prime}(s)$ on $(\varepsilon,t)$ for any $t>0$ and $0<\varepsilon<t$, we obtain from (6.5) that

[TABLE]

Since $\lim_{t\to 0^{+}}f(t)=+\infty$, we know that $\lim_{t\to 0^{+}}g(t)=0$. Letting $\varepsilon\to 0^{+}$ in (6.6), we get

[TABLE]

Consequently

[TABLE]

Hence, we conclude that

[TABLE]

The upper bound estimate (1.7) of sub-elliptic Dirichlet heat kernel is proved, where $C$ is the Sobolev constant in (2.4). This completes the proof of Theorem 1.1. ∎

6.2 Proof of Theorem 1.2.

Proof.

Proposition 4.1 gives us the following:

[TABLE]

It follows from Theorem 1.1 that

[TABLE]

Then, combining (6.8) and (6.9), we get

[TABLE]

Integrating (6.10) with respect to $x$ on $\Omega$ and using the fact $\int_{\Omega}\phi_{i}^{2}(x)dx=1$, we obtain

[TABLE]

Since $x\mapsto e^{-x}$ is a convex function, then (6.11) implies that

[TABLE]

Putting $t=\frac{k}{\sum_{i=1}^{k}\lambda_{i}}$ into (6.12), then

[TABLE]

Here $C_{1}=(Ce|\Omega|)^{-\frac{2}{\tilde{\nu}}}$ is a positive constant depending on $\Omega$ and $\tilde{\nu}$.

The proof of the Theorem 1.2 is now complete. ∎

6.3 Proof of Theorem 1.3.

We use the following Tauberian theorem to prove Theorem 1.3.

Proposition 6.1** (Tauberian theorem).**

Suppose that $\{\lambda_{n}\}_{n\in\mathbb{N}}$ is a sequence of positive real numbers, and for every $t>0$ the series

[TABLE]

Then for $r>0$ and $a\in\mathbb{R}$, the following two arguments are equivalent:

[TABLE]

• 2.

[TABLE]

where $N(\lambda)=\#\{n|~{}0<\lambda_{n}\leq\lambda\}$ for $\lambda>0$.

Proof.

See Theorem 1.1 in [2]. ∎

Proof of Theorem 1.3.

From Proposition 5.4, we know that for the sub-elliptic Dirichlet heat kernel $h_{D}(x,y,t)$, there exists a non-negative function $\gamma_{0}$ on $\overline{\Omega}$ such that

[TABLE]

Hence, (6.17) implies

[TABLE]

Let $H=\{x\in\Omega|\nu(x)=\tilde{\nu}\}$ and $\chi_{H}$ be the characteristic function of $H$. We can derive from (6.18) that

[TABLE]

According to Theorem 1.1, we have

[TABLE]

Combining (6.19) and (6.20), it follows from the Lebesgue’s dominant convergence theorem that

[TABLE]

Here $\gamma_{0}(x)>0$ for any $x\in H=\{x\in\Omega|~{}\nu(x)=\tilde{\nu}\}$.

On the other hand, from Proposition 4.1 we get

[TABLE]

It follows from (6.21) and (6.22) that

[TABLE]

Then, by using the Proposition 6.1, we obtain

[TABLE]

where $N(\lambda)=\#\{k|~{}0<\lambda_{k}\leq\lambda\}$.

Taking $\lambda=\lambda_{k}$, since $\lambda_{k}\to+\infty$ as $k\to+\infty$, then (6.24) implies $N(\lambda_{k})=k+o(\lambda_{k}^{\frac{\tilde{\nu}}{2}})$ as $k\to+\infty$. Hence, we can also deduce from (6.24) that

[TABLE]

This straightforward implies that

If $|H|>0$,

[TABLE]

• 2.

If $|H|=0$,

[TABLE]

Theorem 1.3 is proved. ∎

7 Proofs of Theorem 1.4 and Theorem 1.5

7.1 Proof of Theorem 1.4.

We shall use the generalization of an approach in [30] to give the proof of Theorem 1.4. First, we prove the following proposition.

Proposition 7.1**.**

If $X=(X_{1},X_{2},\ldots,X_{m})$ satisfy the assumptions in Theorem 1.4, then for any $\lambda>0$, we have

[TABLE]

where the constant $C>0$ is dependent on $X$ and $\Omega$, $\lambda_{k}$ is the $k^{th}$ Dirichlet eigenvalue of $-\triangle_{X}$ on $\Omega$, $(\lambda-\lambda_{k})_{+}=\lambda-\lambda_{k}$ if $\lambda>\lambda_{k}$ and $(\lambda-\lambda_{k})_{+}=0$ if $\lambda\leq\lambda_{k}$.

Proof of Proposition 7.1.

Let $\phi_{1},\phi_{2},\cdots$ be the orthonormal eigenfunctions of $-\triangle_{X}$ on $\Omega$ which corresponding to the Dirichlet eigenvalues $0<\lambda_{1}<\lambda_{2}\leq\cdots$. It is easy to verify that the functions

[TABLE]

belong to the domain $D(\triangle_{X})$ of operator $\triangle_{X}$. Denote

[TABLE]

Then, if we let $\varphi_{\lambda}(t):=(\lambda-t)_{+}$, we have

[TABLE]

Let $E_{s}$ be the spectral projection of the self-adjoint operator $-\triangle_{X}$. Then we obtain

[TABLE]

Clearly here we have

[TABLE]

Since $\varphi_{\lambda}(t)$ is a convex function, then we use the Jensen inequality to deduce

[TABLE]

A simple calculation gives

[TABLE]

On the other hand, for each $X_{j}=\sum_{k=1}^{n}a_{jk}(x)\partial_{x_{k}}$, we introduce a vector which corresponding to the differential operator $X_{j}$ by

[TABLE]

Then we can deduce that

[TABLE]

where $\left\langle X_{j}I(x),\xi\right\rangle_{\mathbb{R}^{n}}=\sum_{k=1}^{n}a_{jk}(x)\xi_{k}$ is the inner product of vector $X_{j}I(x)$ and $\xi$ in $\mathbb{R}^{n}$. Thus,

[TABLE]

Then, we have

[TABLE]

Recall that $X=(X_{1},X_{2},\cdots,X_{m})$ are $C^{\infty}$ vector fields defined on the compact domain $\overline{\Omega}$, then we have

[TABLE]

where $M=\sup_{x\in\Omega}\left(\sum_{j=1}^{m}|X_{j}I(x)|^{2}\right)<+\infty$. Observe that $\varphi_{\lambda}(t)$ is decrease with respect to $t$, hence we obtain

[TABLE]

where the positive constant $C$ depends on $X$ and $\Omega$. The proof of Proposition 7.1 is complete.

∎

Proof of Theorem 1.4.

Now, we take $\lambda=\lambda_{k}$ in Proposition 7.1. Then we get

[TABLE]

For $k\geq 2$, we have $\lambda_{k}>\lambda_{1}$, this implies $\frac{\lambda_{k}-\lambda_{j}}{\lambda_{k}-\lambda_{1}}\leq 1$ for $j=1,2,\ldots,k-1$. Hence, we have

[TABLE]

Consequently

[TABLE]

The proof of Theorem 1.4 is complete. ∎

7.2 Proof of Theorem 1.5.

Combining Proposition 5.3 with (5.8), we obtain that for Dirichlet heat kernel $h_{D}(x,y,t)$ of sub-elliptic operator $\triangle_{X}=-\sum_{i=1}^{m}X_{i}^{*}X_{i}$, there exists $C_{2}>0$ such that

[TABLE]

holds for all $t\in(0,1),x,y\in\overline{\Omega}$. Here $B_{d_{X^{\prime}}}(x,r)$ is the subunit ball induced by the Carnot-Carathéodory metric $d_{X^{\prime}}(x,y)$ which depends on the extension $X^{\prime}$. In particular, we have

[TABLE]

Integrating (7.5) with respect $x$ on $\Omega$, we obtain

[TABLE]

Now, by using Proposition 2.6, since $\overline{\Omega}$ is a compact subset of $\mathbb{R}^{n}$, there exists $\delta_{0}=\delta_{0}(\overline{\Omega})>0$ and constants $C_{3},C_{4}>0$ such that

[TABLE]

Take $\delta_{1}=\min\{1,\delta_{0}^{2}\}$, by (7.6) and (7.7) we have for a constant $C_{5}>0$

[TABLE]

On the other hand, the formula (2.11) gives

[TABLE]

If the vector fields $X=(X_{1},X_{2},\cdots,X_{m})$ satisfy the condition (A) on $\Omega$, then from (1.13) we have that

[TABLE]

where the second sum in (7.10) is over all $n$-combinations $(Y_{i_{1}},Y_{i_{2}},\cdots,Y_{i_{n}})$ of set $\{X_{j}|1\leq j\leq m\}$. Combining (7.8), (7.9) and (7.10), we get

[TABLE]

where $C_{6}=C_{5}\int_{\Omega}\frac{1}{\sum|\det(Y_{i_{1}},Y_{i_{2}},\cdots,Y_{i_{n}})(x)|}<+\infty$. Recall that the Dirichlet heat kernel $h_{D}(x,y,t)$ can be expanded by the following series which converges uniformly in $\overline{\Omega}\times\overline{\Omega}\times[a,+\infty)$ for any $a>0$,

[TABLE]

From the fact $\int_{\Omega}\phi_{j}^{2}(x)dx=1$, we have for any $k\in\mathbb{N}^{+}$,

[TABLE]

Hence

[TABLE]

Since $x\mapsto e^{-x}$ is a convex function, then from (7.14) we have

[TABLE]

Since $\lambda_{j}\leq\lambda_{j+1}$, then if we take $t=\frac{\delta_{1}}{2}\cdot\frac{k\lambda_{1}}{\sum_{i=1}^{k}\lambda_{i}}\in(0,\delta_{1})$, we can obtain

[TABLE]

Therefore, we can conclude that

[TABLE]

The proof of Theorem 1.5 is complete.

8 Some Examples

In this section, as applications of Theorem 1.2–Theorem 1.5, we give some examples.

Example 8.1** (Kohn Laplacian $\triangle_{\mathbb{H}}$).**

Let $(\mathbb{H}_{n},\circ)$ be the Heisenberg group in $\mathbb{R}^{2n+1}$. Here $\circ$ is the group operation on the Heisenberg group $\mathbb{H}_{n}$ defined as follows:

Given the two points

[TABLE]

and

[TABLE]

Then

[TABLE]

where the point $\cdot$ stands for the inner product in $\mathbb{R}^{n}$.

Consider the Kohn Laplacian on Heisenberg group $\mathbb{H}_{n}\subset\mathbb{R}^{2n+1}$,

[TABLE]

which is induced by the vector fields $X_{j}=\partial_{x_{j}}+2y_{j}\partial_{z},Y_{j}=\partial_{y_{j}}-2x_{j}\partial_{z}$ for $j=1,2,\cdots,n$. In this case, we know the condition (H) and (M) are permissible in $\mathbb{R}^{2n+1}$, with Hörmander index $Q=2$ and Métivier index $\nu=2n+2$.

Let $\Omega\subset\mathbb{R}^{2n+1}$ be a bounded connected open set with non-characteristic smooth boundary for vector fields $X=(X_{1},\cdots,X_{n},Y_{1},\cdots,Y_{n})$. For the Dirichlet eigenvalue problem (1.1) on $\Omega$, Hansson and Laptev [21] have proved that

[TABLE]

where $C_{n}=\sum_{j_{1},\cdots,j_{n}\geq 0}(2(j_{1}+\cdots+j_{n})+n)^{-(n+1)}$.

Now by our estimation in Theorem 1.2, we get

[TABLE]

where $C$ is a positive constant related to $X$ and $\Omega$. Furthermore, we can get an explicit constant $C$ via the comparison of Dirichlet heat kernel and global heat kernel. From the results in [1], we know that $\triangle_{\mathbb{H}}$ has a non-negative global heat kernel $h(x,y,t)$ such that

[TABLE]

where $\xi=(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n},z)\in\mathbb{H}_{n},r^{2}=\sum_{i=1}^{n}(x_{i}^{2}+y_{i}^{2})$. Since the invariance of the operator $\triangle_{\mathbb{H}}$ with respect to left translations, we have

[TABLE]

Moreover, we have that

[TABLE]

where $\alpha_{n}=\int_{0}^{+\infty}\left(\frac{\theta}{\sinh\theta}\right)^{n}d\theta$. Since $h_{D}(x,y,t)\leq h(x,y,t)$, we obtain

[TABLE]

Therefore, for any $t>0$ we can deduce from (8.3) that

[TABLE]

In order to get a sharp constant $C$, we take $t=s\cdot\frac{k}{\sum_{j=1}^{k}\lambda_{j}}$ in (8.4), where $s>0$ is a constant to be determined later. Then, we have

[TABLE]

Now, we let $g(s)=\frac{s}{e^{\frac{s}{n+1}}}$. It is easy to show that $g(n+1)=\max\limits_{s\in(0,+\infty)}g(s)=(n+1)e^{-1}$. Hence, if we put $s=n+1$ in (8.5), we can get a lower bound with an explicit coefficient

[TABLE]

where $\alpha_{n}=\int_{0}^{+\infty}\left(\frac{\theta}{\sinh\theta}\right)^{n}d\theta$.

As we can see that, for the Hörmander vector fields $X$ with $|H|>0$, Theorem 1.2 and Theorem 1.3 give the optimal estimates of Dirichlet eigenvalues. Here we shall give an example below in which the Métivier’s condition (M) will be not satisfied on $\Omega$, but the set $H=\{x\in\Omega~{}|~{}\nu(x)=\tilde{\nu}\}$ has a strict positive measure.

Example 8.2**.**

Let $\Omega\subset\mathbb{R}^{3}$ be a bounded connected open set with smooth boundary $\partial\Omega$ such that $D(2):=\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}||x_{i}|<2,i=1,2,3\}\subset\subset\Omega$. Given the vector fields $X_{1},X_{2},X_{3}$ defined in $\mathbb{R}^{3}$ such that

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

From above assumptions, we can see that the vector fields $X=(X_{1},X_{2},X_{3})$ verify the Hörmander’s condition on $\overline{\Omega}$ with Hömander index $Q=2$. If we denote $H$ by the set

[TABLE]

We know that $H\subset\subset D(2)\subset\subset\Omega$. Then we have

[TABLE]

and

[TABLE]

Therefore

[TABLE]

The vector fields $X=(X_{1},X_{2},X_{3})$ do not satisfy the Métivier’s condition (M), but has generalized Métivier index $\tilde{\nu}$ on $\Omega$, namely

[TABLE]

For the set H=\left\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}\bigg{|}\sqrt{x_{1}^{2}+x_{2}^{2}}\leq\frac{3}{2},-1\leq x_{3}\leq 0\right\}=\{x\in\Omega|\nu(x)=\tilde{\nu}\}, we know that $|H|>0$. If we consider the Dirichlet eigenvalue problem (1.1) for the sub-elliptic operator $\triangle_{X}=-\sum_{j=1}^{3}X_{j}^{*}X_{j}$ on $\Omega$, according to the Theorem 1.2, we get a lower bound for $\lambda_{k}$

[TABLE]

Thus Theorem 1.3 tells us this lower bound is optimal in sense of the order $k$ and there exists $C_{1}>0$ which is dependent on the vector fields $X=(X_{1},X_{2},X_{3})$ and $\Omega$, such that $\lambda_{k}\sim C_{1}k^{\frac{1}{2}}$ as $k\to+\infty$.

In the following example, we shall have $|H|=0$ and the condition (A) is satisfied.

Example 8.3**.**

For $n\geq 3$, let $G=(X_{1},X_{2},\cdots,X_{n-1},Y_{1},Y_{2},\cdots,Y_{n-1})$ be the vector fields defined on an open connected set $W\subset\mathbb{R}^{n}$, the Grushin operator induced by $G$ is defined as follows:

[TABLE]

Assume $\Omega\subset\subset W$ to be a bounded connected open subset of $W$ with smooth boundary $\partial\Omega$ which is non-characteristic for $G$. Also $\Omega$ satisfies that $\Omega\cap\{(0,0,\cdots,0,x_{n})|x_{n}\in\mathbb{R}\}\neq\varnothing$, and the generalized Métivier index $\tilde{\nu}=n+1$. If we let $x^{\prime}=(x_{1},\cdots,x_{n-1})$ then $dx=dx^{\prime}dx_{n}$ and $\Omega\subset\Omega_{x^{\prime}}\times\Omega_{x_{n}}$. Here $\Omega_{x^{\prime}},\Omega_{x_{n}}$ are the projections of $\Omega$ in $\{(x_{1},\cdots,x_{n-1},0)\}$ and $\{(0,\cdots,0,x_{n})\}$. Recall $\Omega\cap\{(\mathbf{0},x_{n})\in\mathbb{R}^{n}\}\neq\varnothing$, that implies $\mathbf{0}\in\Omega_{x^{\prime}}$. Since $\Omega_{x^{\prime}}$ is an open set, there exists $\delta>0$ such that $B(\mathbf{0},\delta)\subset\Omega_{x^{\prime}}$. By a direct calculation, we know that

[TABLE]

where the sum is over all $n$-combinations $(Y_{i_{1}},Y_{i_{2}},\cdots,Y_{i_{n}})$ of the set $\{X_{j},Y_{j}|1\leq j\leq n-1\}$. Observe that

[TABLE]

Thus, in order to verify the assumption (A), it suffices to prove the convergence of integral $\int_{\Omega_{x^{\prime}}}\frac{dx^{\prime}}{|x^{\prime}|}$. Indeed, we can obtain that

[TABLE]

We know that the second part in (8.9) is finite. Then, for the first part in (8.9), we have

[TABLE]

Hence, by Theorem 1.5, we have the following estimate for $\lambda_{k}$

[TABLE]

where $C>0$ is some constant which depends on the vector fields $G$ and $\Omega$.

From the upper bound estimate of $\lambda_{k}$ in Theorem 1.4 and the lower bound estimate (8.10), we know that $\lambda_{k}\approx k^{\frac{2}{n}}$ as $k\to+\infty$ in this example, which indeed improves the results for this Grushin type sub-elliptic operator in [13] and [14].

Finally, we give an example for Grushin type vector fields, in which $|H|=0$ but the condition (A) is not satisfied. In this case, we can see the increase order of $k$ for $\lambda_{k}$ may smaller than $k^{\frac{2}{n}}$.

Example 8.4**.**

For $n=2$, let the Grushin vector fields $X=(\partial_{x_{1}},x_{1}\partial_{x_{2}})$ defined on an open connected set $W\subset\mathbb{R}^{2}$. The Grushin operator induced by $X$ is defined as

[TABLE]

Assume $\Omega\subset\subset W$ to be a bounded connected open subset of $W$ which has smooth and non-characteristic boundary for $X$. Meanwhile $\Omega$ satisfies that $\Omega\cap\{(0,x_{2})|x_{2}\in\mathbb{R}\}\neq\varnothing$. Thus, there exists a point $(0,y_{0})\in\Omega$. Since $\Omega$ is an open set, then we can find $\delta>0$ such that $B((0,y_{0}),\delta)\subset\Omega$. It is easy to get that

[TABLE]

where the sum is over all 2-combinations $(Y_{i_{1}},Y_{i_{2}})$ of set $\{\partial_{x_{1}},x_{1}\partial_{x_{2}}\}$. Observe that

[TABLE]

Therefore, the vector fields do not satisfy the condition (A). However, by calculating directly, we have

[TABLE]

From (7.8), we obtain

[TABLE]

where $C_{7}=\frac{1}{2}C_{5}>0$. Since $\Omega\subset[a,b]\times\Omega_{x_{2}}$ for some $a<0<b$, we can deduce that

[TABLE]

Therefore, we have

[TABLE]

holds for some $0<t<\delta_{1}$. Observe that if we take $t=\frac{1}{\lambda_{k}}$, then there exists $j_{0}\in\mathbb{N}^{+}$ large enough, such that $t<\delta_{1}$ for $k\geq j_{0}$. Thus, we have

[TABLE]

That means $\lambda_{k}\geq Ck(\log k)^{-1}>Ck^{\frac{2}{3}}$ for $k$ large enough. Here the generalized Métivier index $\tilde{\nu}=3$.

9 A remark on uniform the Hörmander condition

In this part, we introduce the uniform version of Hörmander’s condition which was defined in [27] and [28].

For the vector fields $X^{\prime}=(Z_{1},Z_{2},\cdots,Z_{q})$ defined in $\mathbb{R}^{n}$, we denote $J=(j_{1},\cdots,j_{k})$ with $1\leq j_{i}\leq m$, $|J|=k$ is the length of $J$. Then the $k$ order commutator $Z_{J}$ is defined as

[TABLE]

We say that the vector fields $X^{\prime}=(Z_{1},Z_{2},\cdots,Z_{q})$ satisfy the uniform version of Hörmander’s condition in $\mathbb{R}^{n}$ if there exists a positive integer $Q$ and a positive constant $\alpha$ such that

[TABLE]

Here $\left\langle\cdot,\cdot\right\rangle_{\mathbb{R}^{n}}$ is the inner product in $\mathbb{R}^{n}$, $Z_{J}I(x)$ is the vector in $\mathbb{R}^{n}$ which corresponding to the differential operator $Z_{J}$.

Now, we assume that $X^{\prime}=(Z_{1},Z_{2},\cdots,Z_{q})$ is an extension of $X=(X_{1},X_{2},\cdots,X_{m})$ in Proposition 5.1 and satisfies the Hörmander’s condition (H) in $\mathbb{R}^{n}$ with Hörmander’s index $Q$. Moreover, we know that

[TABLE]

Then we have

Proposition 9.1**.**

The vector fields $X^{\prime}$ in (9.2) satisfies the uniform Hörmander’s condition (9.1) in $\mathbb{R}^{n}$ for the positive integer $Q$ and some constant $\alpha>0$.

Proof.

It is simple to see that for any $x\in\mathbb{R}^{n}\setminus\Omega_{2}$ and $\eta\in\mathbb{S}^{n-1}$, we have

[TABLE]

Therefore, it suffices to prove that

[TABLE]

holds for some $\alpha>0$. If the assertion would not hold, then for any $n\in\mathbb{N}$, there exists a sequence $\{x_{n}\}_{n=1}^{\infty}\subset\overline{\Omega_{2}}$ such that

[TABLE]

Hence, we can find a sequence $\{\eta_{n}\}_{n=1}^{\infty}\subset\mathbb{S}^{n-1}$ such that

[TABLE]

Since $(x_{n},\eta_{n})\in\overline{\Omega_{2}}\times\mathbb{S}^{n-1}$ and $\overline{\Omega_{2}}\times\mathbb{S}^{n-1}$ is a compact set, we can find a subsequence $(x_{n_{k}},\eta_{n_{k}})\to(x_{0},\eta_{0})\in\overline{\Omega_{2}}\times\mathbb{S}^{n-1}$ as $k\to+\infty$. Thus, we can deduce from (9.4) that

[TABLE]

Now, let $Y_{j}=\sum_{k=1}^{n}a_{jk}(x)\partial_{x_{k}}(1\leq j\leq n)$ be arbitrary $n$ vector fields which are chosen from the set $\{Z_{J}||J|\leq Q\}$. It can be deduced from (9.5) that

[TABLE]

Therefore, (9.6) implies $\det(Y_{1},Y_{2},\cdots,Y_{n})(x_{0})=0$, which means $Z_{1},Z_{2},\cdots,Z_{q}$ together with their commutators up to length $Q$ cannot span the tangent space $T_{x_{0}}(\mathbb{R}^{n})$ at the point $x_{0}$. This leads to a contradiction. Thus we have the conclusion of Proposition 9.1. ∎

Acknowledgments

The first version of this paper was done when the first author visited the Max-Planck Institute for Mathematics in the Sciences, Leipzig during July-August of 2018 as a visiting professor. He would like to thank Professor J. Jost (Max-Planck Institute for Mathematics in the Sciences, Leipzig) for the invitation and financial support.