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


Then
$$
\partial\_t g(t, x) = \Delta g(t, x) = \left(\frac{|x|^2-2 d t}{4 t^2}\right) g(t, x).
$$


In a previous [thread](https://mathoverflow.net/questions/448491/is-there-f-such-that-int-0t-fs-mathrm-d-s-infty-and-partial-t-g), I got that



> 
> There do *not* exist a constant $C \ge 1$ and a measurable function $f:(0, \infty) \to \mathbb R\_+$ such that
> 
> 
> * $\int\_0^t f(s) \, \mathrm d s < +\infty$ for all $t>0$.
> * $|\partial\_t g| (t, x) \le f(t)g(Ct, x)$ for all $t>0$ and $x \in \mathbb R^d$.
> 
> 
> 


I would like to ask if the following relaxation holds, i.e.,



> 
> There exist measurable functions $f, h:(0, \infty) \to \mathbb R\_+$ such that
> 
> 
> * $h$ is Lipschitz.
> * $\int\_0^t f(s) \, \mathrm d s < +\infty$ for all $t>0$.
> * $|\partial\_t g| (t, x) \le f(t)g(h(t), x)$ for all $t>0$ and $x \in \mathbb R^d$.
> 
> 
> 


Thank you so much for your help!



Are there $f,h$ such that $h$ is Lipschitz, $\int_0^t f(s)\,\mathrm d s<\infty$ and $|\partial_t g| (t, x) \le f(t)g(h(t), x)$?
This is impossible for **any** positive function $h$. Indeed, for $|x|=\sqrt t$, the inequality $|\partial\_t g| (t, x) \le f(t)g(h(t), x)$ implies
$$f(t)\ge\frac Ct\,u^{-d/2}e^{cu}\ge\frac Bt$$
for all real $t>0$, where $C,c,B$ are certain positive real numbers not depending on $t$, and $u:=t/h(t)$. So, $\int\_0^t f=\infty$ for all real $t>0$.


