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 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!