Let $g$ be the heat kernel. Are there constants $C_1, C_2>0$ such that $\frac{g(t_1, \cdot)}{t_1} \le C_1 \frac{g(C_2 t_2, \cdot)}{\sqrt{t_2}}$?

Question

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!

Answer 1

The answer is no. E.g., let $t_1\sim t_2\downarrow 0$ and $|x|\sim\sqrt{t_2}$.