Let $\Omega$ be a bounded open domain in $\mathbb{R}^{n+1}$ with smooth boundary. $\Omega\cap\{(0,\cdots,0,y)\in\mathbb{R}^{n+1}| y\in \mathbb{R}\}\neq \varnothing$
Consider the sub-elliptic operator $$ L=\frac{1}{2}\left(\sum_{k=1}^{n}X_{i}^2+Y^2 \right)=\frac{1}{2}\sum_{i=1}^{n}\partial_{x_{i}}^2+2|x|^2\partial_{y}^2$$ where $X_{k}=\partial_{x_{k}},Y=2|x|\partial_{y}$ are vector fields on $\Omega$. We know the heat kernel for the sub-elliptic operator $L_{n}$ is given by $$ K(x_{0},x,t)=\frac{1}{(2\pi t)^{1+\frac{n}{2}}}\int_{-\infty}^{+\infty} \left(\frac{2s}{\sinh(2s)}\right)^{\frac{n}{2}}\exp\left(\frac{i(y-y_{0})-s(|x|^2+|x_{0}|^2)\coth(2s)+\frac{2s}{\sinh(2s)}<x,x_{0}>}{t}\right)ds $$ Now, for $x_{0}=x,y_{0}=y$.$P=(x,y)$ is a point in $\Omega$. the heat kernel $K$ become $$ K(P,P,t)=\frac{1}{(2\pi t)^{1+\frac{n}{2}}}\int_{-\infty}^{+\infty}\left(\frac{2s}{\sinh(2s)}\right)^{\frac{n}{2}}\exp\left(\frac{-2s|x|^2\coth(2s)+\frac{2s|x|^2}{\sinh(2s)}}{t} \right)ds $$ For $0<t\leq 1$. From Andrew L. Ursitti's article <Spectral Asymptotics for Operators of Hormander Type> there exist a constant $C_{L}$ that $$ K(P,P,t)\leq \frac{C_{L}}{t^{\frac{n+2}{2}}}\qquad \forall P\in\Omega $$ My Question is: Can we get the value of $C_{L}$ for the sub-elliptic operator $L$ ? In another words, shall we find and compute a constant $C=C(L,\Omega)$ that depend on $L$ and $\Omega$ such that $$ \int_{-\infty}^{+\infty}\left(\frac{2s}{\sinh(2s)}\right)^{\frac{n}{2}}\exp\left(\frac{-2s|x|^2\coth(2s)+\frac{2s|x|^2}{\sinh(2s)}}{t} \right)ds\leq C $$ ?
