Given $f : \mathbf{R}^d \to \mathbf{R}$, define $P_t f$ by
\begin{align} (P_t f)(x) = \mathbf{E} \left[ f (x + \sqrt{t} G) \right], \end{align}
where $G \sim \mathcal{N} (0, I_d)$ is a standard Gaussian random variable.
I am interested in statements of the form
If $f$ (or one of its derivatives of some order) has modulus of continuity $\omega$, then the derivatives (of some order) of $P_t f$ can be bounded uniformly as $\| \nabla^k P_t f \| \leqslant B(k, t, \omega, d)$, where the function $B$ is relatively explicit.
The term $\| \nabla^k P_t f \|$ should be interpreted as taking the absolute supremum in $x$ for $k = 0$, supremum in $x$ of the Euclidean norm for $k = 1$, supremum in $x$ of the operator norm for $k = 2$, and so on. Naturally, I am basically happy to work with polynomial $\omega$ (i.e. Hoelder-type regularity assumptions), but a more general treatment is also of interest.
Anyways, I can prove some bounds of this form by hand using probabilistic methods and more-or-less well-known techniques which boil down to integration by parts. These approaches come naturally enough to me because of my background, but given the wealth of perspectives which one can take on the heat flow, it seems likely that there are plenty of other equally useful approaches with which one could have success (e.g. I have not really played with Fourier-type approaches, or non-probabilistic PDE strategies).
Instead of asking for specific bounds, I am most interested in identifying references which can deliver a relatively thorough treatment of this problem. It seems to me that this problem should be quite well-understood, and so it appears that the most efficient solution (for both my specific problem and for my general understanding) could be to find an appropriate reference. Papers and books are both equally acceptable for me.
One minor point (which may be a more modern consideration) is that I insist on bounds which track the dependence upon the ambient dimension, or at least upon techniques which enable such an analysis.
