We consider the heat kernel
$$
g :\mathbb R\_{>0} \times \mathbb R^d \to \mathbb R,\quad (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$


Let $0 < t\_1 < t\_2 <\infty$ and $1\le\lambda<\infty$ such that $\frac{t\_2}{t\_1} \le \lambda$. I would like to ask if there are constants $C\_1, C\_2>0$ (depending only on $\lambda$) such that
$$
\frac{g(t\_1, x)}{t\_1} \le C\_1 \frac{g(C\_2 t\_2, x)}{\sqrt{t\_2}}
\quad \forall x \in \mathbb R^d.
$$


Thank you so much for your elaboration!



Let $g$ be the heat kernel. Are there constants $C_1, C_2>0$ such that $\frac{g(t_1, \cdot)}{t_1} \le C_1 \frac{g(C_2 t_2, \cdot)}{\sqrt{t_2}}$?
The answer is no. E.g., let $t\_1\sim t\_2\downarrow 0$ and $|x|\sim\sqrt{t\_2}$.


