Let the heat kernel on $(0,\infty)\times \mathbb R^n$ be given by
$\Psi(x,t) = (4\pi t )^{-\frac{n}{2}} e^{ -\dfrac{|x|^2}{4t} }$
for $t>0$, otherwise $0$ except at the origin of space-time.
It is clear that the heat kernel is smooth everywhere except at the origin. Let $\delta > 0$. How can you show that ( the absolute values of ) the heat kernel and all of its derivatives are uniformly bounded on $[\delta, \infty ) \times \mathbb R^n$, say, by some constant $C_\delta > 0$?
It seems clear as each derivative does not change the relation between the exponents in $x$ and $t$. However, I am stuck and don't know how to proceed. Search on the web provided me only with statements of this fact without a proof.
Thank you.