let $\mathbb{H}$ be the hyperbolic plane and let $k(t,x,y)$ be the associated heat kernel.


I am wondering, if for any fixed $y\in M$ and $\epsilon >0$ the function $u\_t(x):=k(t,x,y)$ is continuous in $S:=\lbrace x\in M : d(x,y)>\epsilon \rbrace$ uniformly in $t\in(0,\infty)$?


For instance in Euclidean space $\mathbb{R}^n$ the heat kernel is given by $k(t,x,y)=\frac{1}{(4\pi t)^{n/2}}e^{-\frac{\Vert x-y \Vert^2}{4t}}$. Hence for $\mathbb{R}^n$ the claim is true because the function $u\_t(x)$ has bounded derivative on S, i.e. $\sup\_{t>0} \vert \nabla u\_t(x) \vert\leq C$.


There is a formula for hyperbolic plane as well, given by


$$k(t,x,y)=\frac{\sqrt{2}}{(4\pi t)^{3/2}}e^{-\frac{t}{4}}\int\limits\_{d(x,y)}^{\infty}\frac{s e^{-\frac{s^2}{4t}}}{\sqrt{\cosh{s}-\cosh{d(x,y)}}}ds$$
but I do not know how to estimate the derivative for this function.


Q: How can I prove/disprove that for any $x^{\*}\in\mathbb{H}^2, x^{\*}\neq y, \forall \epsilon>0$ $\exists\delta>0,$ s.t. $\vert u\_t(x)-u\_t(x^{\*}) \vert\leq \epsilon$ for all $x\in \mathbb{H^2}$ with $d(x,x^{\*})<\delta$? If the above statement is wrong, What happens if we restrict the domain to $t\in (0,T)$ for $T>0$ arbitrary and fixed?


Any help will be very appreciated!


Best wishes 



Is the heat kernel for the hyperbolic plane uniformly continuous in $t\in(0,\infty)$?
It follows from the interior Schauder estimates for parabolic equations. Namely, put for simplicity's sake $y=0$, $k(t,x)=k(t,x,0)$ and for negative values of $t$ continue $k$ by zero: $k(x,t)=0$, $t<0.$ For $r>0$ denote $B\_r(x\_0)=\{x\in \mathbb{H}^2|d(x,x\_0)<r\}$ a ball on the plane and $B\_r=B\_r(0)$. Then $k(x,t)$ is bounded on a cylindrical domain $H\_r=\mathbb R\times (\mathbb{H}^2\backslash B\_r)$. Now fixing $\alpha\in(0,1)$ and applying the interior Schauder estimate in the cylinder $G\_r(x\_0,t\_0)=(t\_0-r^2,t\_0)\times B\_r(x\_0)$, $d(x\_0,0)\ge 2r$, we have
$$
\|k\|\_{C^{2,\alpha}(G\_{r/2}(x\_0,t\_0))}\le C\_r\sup\_{G\_{r}(x\_0,t\_0)}|k|\le C\_r\sup\_{H\_r}|k|.
$$
It follows that 
$$
\sup\_{H\_{2r}}|\nabla\_x k|\le C\_r\sup\_{H\_r}|k|.
$$


