# Prove the positivity of the subelliptic operator heat kernel

Let $X=(X_{1},X_{2},\cdots,X_{m})$ be $C^{\infty}$ real vector fields defined in $\mathbb{R}^{n} (n>2)$ satisfying the Hormander's condition at every point $x\in\mathbb{R}^{n}$, Let $W$ be a bounded open set and $$H_{X}^{1}(W)=\{u\in L^{2}(W)|X_{j}u\in L^{2}(W), j=1,\cdots,m\}.$$ Let open connected set $\Omega\subset W$, we define $H_{X,0}^{1}(\Omega)$ to be the closure of $C_{0}^{\infty}(\Omega)$ in $H_{X}^{1}(W)$, Consider the Dirichlet heat kernel of $\triangle_{X}=\sum_{i=1}^{m}X_{i}^2$ on $\Omega$, Since the eigenfunctions $\{\phi_{i}\}$ is the orthogonal basis in $L^2(\Omega)$, I use series to define the Dirichlet heat kernel: $$h(t,x,y)=\sum_{i=1}^{\infty}e^{-\lambda_{i}t}\phi_{i}(x)\phi_{i}(y)$$ Then I get that $\|\phi_{i}\|_{\infty}\leq C\lambda_{i}^{\frac{Q}{4}}$, where $Q$ is the homogeneous dimension of $\Omega$, that makes the series converge to $h(t,x,y)$ uniform in $[a,+\infty)\times\Omega\times\Omega$, for any $a>0$, and or any $f_{0}\in L^2(\Omega)$, the function give by $$f(x,t)=\int_{\Omega}h(x,y,t)f_{0}(y)dy$$ solves the heat equation \begin{equation*} \left(\triangle_{X}-\frac{\partial}{\partial t}\right)f(x,t)=0 \quad\text{on}\quad \overline{\Omega}\times(0,\infty) \end{equation*} and \begin{equation*} f(x,t)=0 \qquad \text{on} \qquad \partial\Omega\times(0,\infty) \end{equation*} with the initial condition \begin{equation*} \lim_{t\to 0}f(x,t)=f_{0}(x) ~~\text{in}~~ L^2(\Omega) \end{equation*} also we can prove $$\lim_{t\to 0}h(x,y,t)=\delta_{x}(y)$$ in distribution sense. and $$h(x,y,t+s)=\int_{\Omega}h(x,z,t)h(z,y,t)dz$$ But I don't know how to prove positiviy of $h(x,y,t)$: $$h(x,y,t)>0 ~~~ \forall x\in\Omega, y\in\Omega, t>0$$ and $$\int_{\Omega}h(x,y,t)dy\leq 1$$ because the degenerate of the vector fields $X$, I tried with Bony Bony's article strong maximum principle to prove it, but I need $f(x,t)\in C([0,T]\times\overline\Omega)$ when $f_{0}\in C(\overline\Omega)$, the $f(x,t)$ converge $f_{0}$ in $L^2$ can't show that, is there exist some regularity of solution $f(x,t)$ rely on $f_{0}(y)$? Can someone help me ? Thanks a lot!