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


*Corollary 1.3* and *Theorem 1.2* in the paper [Upper Bounds of Derivatives of the Heat Kernel on an Arbitrary Complete Manifold](https://www.sciencedirect.com/science/article/pii/S0022123685710166) by Alexander Grigor'yan imply there is a constant $C$ such that
$$
|\partial\_t g| (t, x) \le C\frac{g(2t, x)}{t}
\quad \forall t>0, \forall x \in \mathbb R^d.
$$


This upper bound is not good enough for my purpose because $\int\_0^t \frac{\mathrm d s}{s} = +\infty$ for any $t>0$. I would like to ask if the following improvement is possible, i.e.,



> 
> There 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 $x \in \mathbb R^d$.
> 
> 
> 


Any reference is appreciated. Thank you so much for your help!



Is there $f$ such that $\int_0^t f(s)\,\mathrm d s<\infty$ and $|\partial_t g| (t, x) \le f(t)g(Ct, x)$ for all $t>0$ and $x \in \mathbb R^d$?
Such a function $f$ does not exist.


Indeed, if the inequality
$|\partial\_t g|(t,x)\le f(t)g(Ct,x)$ holds for all $t>0$ and $x\in\mathbb R^d$, then it holds for $x=0$, so that
$$f(t)\ge f\_\*(t):=\frac{|\partial\_t g|(t,0)}{g(Ct,0)}
=C^{d/2}\frac{d }{2 t}$$
for all real $t>0$.
So, $\int\_0^t f\ge\int\_0^t f\_\*=\infty$ for all $t>0$.


(Instead of choosing $x=0$, here we can more generally let $|x|=O(\sqrt t)$.)


