Let $\Omega$ be a bounded open domain in $R^{n}$ with smooth boundary and $X=(X_{1},X_{2},\cdots,X_{m})$ be a system of real smooth vector fields defined on $\Omega\subset \mathbb{R}^{n}$. If $X$ satisfy the Hormander condition, we have some result about the heat kernel of $L=\sum_{i=1}^{m} X_{i}^2$,like Gerard ben Arous's article <Developpement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale>,but is there exist some heat kernel result for $L^{m} (m\in N)$ ? For some classical opeartor,like Kohn Laplacian $\Delta_{\mathbf{H}}$ on Hessionberg group $\Omega\subset R^{2N+1}$, we know the heat kernel is $$ H(0,x,t)=\frac{1}{(2\pi t)^{n+1}}\int \left(\frac{2s}{\sinh(2s)} \right)^{n}\cdot e^{-\frac{f(x,t,s)}{t}}ds $$ Where $f(x,t,s)=-i(y)s+s|x|^2\coth(2s)$. (See Ovdiu Calin's book <Heat Kernels for Elliptic and Sub-elliptic Operators>),but can we compute the heat kernel for $(\Delta_{\mathbb{H}})^{m}$ ? If $X=(X_{1},X_{2},\cdots,X_{m})$ not satisfy the Hormander condition, but satisfy the following logarithmic regularity estimate (See Y.Morimoto, C.J. Xu's article <Logarithmic Sobolev inequality and semi-linear Dirichlet problems for infinitely degenerate elliptic operators> ) $$ \|(\log \Lambda)^{s}u\|^{2}_{L^{2}(\Omega)}\leq C_{0}\left[\int_{\Omega}|Xu|^2 dx+\|u\|^{2}_{L^{2}(\Omega)}\right] $$ for all $u\in C_{0}^{\infty}(\Omega)$, where $s>1$, $C_{0}>0$ and $\Lambda=(e^2+|\nabla|^{2})^{\frac{1}{2}}.$ we know the $L$ is a sub-elliptic operator. Is there exist some result about the heat kernel for operator $L=\sum_{i=1}^{m} X_{i}^2 $ ? or even more $L^m$ ?

More precisely,for some operator's like $L=\sum_{i=1}^{m} X_{i}^2$. where $X=(X_{1},X_{2},\cdots,X_{n})=(\partial_{x_1},\cdots,\partial_{x_{n-1}},\phi(x_1)\partial_{x_n})$, for $n\geq 2$, where $s>0$ and $$ \phi(x_1)=\left\{ \begin{array}{ll} \exp\left(-\frac{1}{|x_{1}\sin\left(\frac{\pi}{x_{1}}\right)|^{\frac{1}{s}}}\right), & \hbox{$x_{1}\neq 0$;} \\ 0, & \hbox{$x_{1}=0$.} \end{array} \right.$$ We know the $X$ is infinitely degenerate on $\Gamma=\bigcup_{j\in \mathbb{Z}_{+}}\Gamma_{j}$, for $\Gamma_{j}=\{x_{1}=\frac{1}{j}\},j\geq 1$, and $\Gamma_{0}=\{x_{1}=0 \}.$ I wonder there is a method to compute it's heat kernel?


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