Let $\Omega\subset\mathbb R^n$ be a neighborhood of $0$, and denote by $i$ the square root of $-1$. We consider the following system of complex vector fields

$ \begin{equation}\label{syst} {\mathcal P}_j=\partial_{x_j}-i \left( {\partial_{x_j}\varphi (x)} \right)\partial_{t}, \quad j=1, \cdots, n, \quad (x, t)\in\Omega\times\mathbb R, \end{equation} $

(1.1)

where $\varphi(x)$ is a real-valued function defined in $\Omega$. This system was first [4], and considered therein is more general case for $t$ varies in $\mathbb R^m$ rather than $\mathbb R.$ Denote by $(\xi, \tau)$ the dual variables of $(x, t)$. Then the principle symbol $\sigma$ for the system $\left\{ {\mathcal P}_j \right\}_{1\leq j\leq n}$ is

$ \begin{eqnarray*} \sigma (x, t; \xi, \tau)=\left( {i \xi_1+\left( {\partial_{x_1}\varphi } \right)\tau, \cdots, i \xi_j+\left( {\partial_{x_j}\varphi } \right)\tau} \right)\in \mathbb C^n \end{eqnarray*} $

with $ \left( {x, t; \xi, \tau} \right)\in T^*\left( {\Omega\times\mathbb R_t} \right)\setminus\left\{ 0 \right\}, $ and thus the characteristic set is

$ \begin{eqnarray*} \big\{(x, t;\xi, \tau)\in T^*\left( {\Omega\times\mathbb R_t} \right)\setminus\left\{ 0 \right\}~\big|~\xi=0, ~\tau\neq 0, ~\nabla\varphi(x)=0\big\}. \end{eqnarray*} $

Since outside the characteristic set the system $\left\{ {\mathcal P}_j \right\}_{1\leq j\leq n}$ is (microlocally) elliptic, we only need to study the microlocal hypoellipticity in the two components $\left\{ \tau>0 \right\}$ and $\left\{ \tau<0 \right\}$ under the assumption that

$ \begin{eqnarray}\label{gra} \nabla \varphi(0)=0. \end{eqnarray} $

(1.2)

Note we may assume $\varphi(0)=0$ if replacing $\varphi$ by $\varphi-\varphi(0).$ Throughout the paper we will always suppose $\varphi$ satisfies the following condition of finite type

$ \begin{equation}\label{finite} \sum\limits_{1\leq \left| { \alpha } \right| \leq k-1}|\partial^\alpha \varphi(0)|=0\quad {\rm and}\quad \sum\limits_{\left| { \alpha } \right|=k}|\partial^\alpha \varphi(0)|>0. \end{equation} $

(1.3)

for some positive integer $k.$ In view of (1.2) it suffices to consider the nontrivial case of $k\geq 2$ for the maximal hypoellipticity. By maximal hypoellipticity (in the sense of Helffer-Nourrigat [2]), it means the existence of a neighborhood $\tilde\Omega\subset \Omega $ of $0$ and a constant $C$ such that for any $u\in C_0^\infty(\tilde\Omega\times \mathbb R), $ we have

$ \begin{equation}\label{maxhyp} ||\partial_{x}u||_{L^2(\mathbb R^{n+1})} +||\left( {\partial_{x}\varphi} \right) \partial_{t}u||_{L^2(\mathbb R^{n+1})} \leq C \Big(\sum\limits_{j=1}^n||{\mathcal P}_{j}u||_{L^2(\mathbb R^{n+1})} + ||u||_{L^2(\mathbb R^{n+1})}\Big), \end{equation} $

(1.4)

where and throughout paper we use the notation $ ||\vec a||_{L^2}=\Big(\sum\limits_{1\leq j\leq n}||a_j||_{L^2}^2 \Big)^{1/2} $ for vector-valued functions $\vec a=(a_1, \cdots, a_n).$ Note that the maximal hypoellipticity along with the condition (1.3) yields the following subellptic estimate

$ \begin{eqnarray*} ||\partial_{x}u||_{L^2(\mathbb R^{n+1})}+|||D_t|^{1\over k} u||_{L^2(\mathbb R^{n+1})} \leq C \Big(\sum\limits_{j=1}^n||{\mathcal P}_{j}u||_{L^2(\mathbb R^{n+1})} + ||u||_{L^2(\mathbb R^{n+1})}\Big). \end{eqnarray*} $

Thus the subellipticity is in some sense intermediate between the maximal hypoellipticity and the local hypoellipticity.

Observe the system $\left\{ {\mathcal P}_j \right\}_{1\leq j\leq n} $ is translation invariant for $t$. So we may perform partial Fourier transform with respect to $t$, and study the maximal microhypoellipticity, in the two directions $\tau>0$ and $\tau<0.$ Indeed we only need consider without loss of generality the maximal microhypoellipticity in positive direction $\tau>0, $ since the other direction $\tau<0$ can be treated similarly by replacing $\varphi$ by $-\varphi.$ Consider the resulting system as follows after taking partial Fourier transform for $t\in\mathbb R.$

$ \begin{equation}\label{system2} \partial_{x_j}+\tau \left( {\partial_{x_j}\varphi} \right), \quad j=1, \cdots, n, \quad x\in\Omega\subset\mathbb R^n, \end{equation} $

(1.5)

and we will show the maximal microhypoellipticity at $0$ in the positive direction in $\tau>0$, which means the existence of a positive number $\tau_0>0, $ a constant $C>0$ and a neighborhood $\tilde\Omega\subset \Omega$ of $0$ such that

$ \begin{array}{l} \forall~ \tau\geq \tau_0, ~\forall~u\in C_0^\infty(\tilde\Omega), \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; ||\partial_{x}u||^2_{L^2} +||\tau \left( {\partial_{x}\varphi} \right) u||^2_{L^2}\leq C\Big( || \partial_{x}+\tau \left( {\partial_{x}\varphi} \right)u||^2_{L^2} +||u||^2_{L^2}\Big), \end{array} $

(1.6)

where and throughout the paper we denote $||\cdot||_{L^2(\mathbb R^n)}$ by $||\cdot||_{L^2}$ for short if no confusion occurs. We remark the operators defined in (1.5) is closely related to the semi-classical Witten Laplacian $ \triangle_{ \tau V}^{(0)} =-\triangle_x +\tau^{2} |\partial_x V|^2-\tau \triangle_x V $ with $\tau^{-1}$ the semi-classical parameter, by the relationship

$ \begin{eqnarray*} || \partial_{x}+\tau \left( {\partial_{x}\varphi} \right)u||^2_{L^2}=\left( {\triangle_{ \tau V}^{(0)} u, \ u} \right)_{L^2}, \end{eqnarray*} $

where $\left( {\cdot, \cdot} \right)_{L^2}$ stands for the inner product in $L^2(\mathbb R^n).$ Helffer-Nier [1] conjectured $ \triangle_{ \tau V}^{(0)} $ is subelliptic near $0$ if $\varphi$ is analytic and has no local minimum near $0, $ and this still remains open so far. Note (1.6) is a local estimate concerning the sharp regularity near $0\in \mathbb R^n$ for $\tau>0$, and we have also its global counterpart, which is of independent interest for analyzing the spectral property of the resolvent and the semi-classical lower bound of Witten Laplacian. We refer to Helffer-Nier's work [1] for the detailed presentation on the topic of global maximal hypoellipticity and its application to the spectral analysis on Witten Laplacian.

Theorem 1.1 (Maximal microhypoellipticity for $\tau>0$) Let $\varphi$ be a polynomial satisfying condition (1.3) with $k\geq 2$. Denote by $\lambda_j , 1\leq j\leq n, $ the eigenvalues of the Hessian matrix $(\partial_{x_i}\partial_{x_j}\varphi)_{n\times n}.$ Suppose there exists a constant $C_*>0$ such that for any $ x\in \Omega, $ we have the following estimates: if $k=2, $ then

$ \begin{equation}\label{11090401+} \sum\limits_{\lambda_j(x)>0}\lambda_j(x) \leq C_*\Big(\sum\limits_{\lambda_j(x)<0}|\lambda_j(x)|+ |\partial_x\varphi(x)|^{\epsilon_0} \Big), \end{equation} $

(1.7)

and if $k>2, $ then

$ \begin{equation}\label{11090401} \sum\limits_{\lambda_j(x)>0}\lambda_j(x) \leq C_*\Big(\sum\limits_{\lambda_j(x)<0}|\lambda_j(x)|+ |\partial_x\varphi(x)|^{\frac{k-2}{k-1}}+\sum\limits_{2\leq\left| { \beta } \right|\leq k-1}\big|\partial_x^\beta\varphi(x)\big|^{\mu_\beta}\Big), \end{equation} $

(1.8)

where $\epsilon_0>0$ is an arbitrarily small number and $\mu_\beta$ are given numbers with $\mu_\beta>(k-2)/(k-\left| { \beta } \right|)$ for $2\leq \left| { \beta } \right|\leq k-1.$ Then the system ${\mathcal P}_j$ defined in (1.1) is maximally microhypoelliptic in positive position $\tau>0$, that is, estimate (1.6) holds.

Replacing $\varphi$ by $-\varphi$ we can get the maximal microhypoellipticity for $\tau<0$, and thus the maximal hypoellipticity for all $\tau$.

Corollary 1.2 (Maximal hypoellipticity) Under the same assumption as Theorem 1.1 with (1.7) and (1.8) replaced by the estimate that for any $x\in \Omega, $

$ \begin{equation*} \sum\limits_{j=1}^n |\lambda_j(x)| \leq \left\{ \begin{aligned} & C_* |\partial_x\varphi(x)|^{\epsilon_0}, \quad {\rm if}~k=2, \\[4pt] & C_* \Big( |\partial_x\varphi(x)|^{\frac{k-2}{k-1}}+\sum\limits_{2\leq\left| { \beta } \right|\leq k-1}\big|\partial_x^\beta\varphi(x)\big|^{\mu_\beta}\Big), \quad {\rm if}~k>2, \end{aligned} \right. \end{equation*} $

the system ${\mathcal P}_j$ defined in (1.1) is maximally hypoelliptic, that is, estimate (1.4) holds.

Remark 1.3 We need only verify conditions (1.7) and (1.8) for these points where $\Delta\varphi$ is positive, since it obviously holds for the points where $\Delta\varphi\leq 0.$

The details of the proof for the main result were given by [3].