Is there anything in the literature related to obtaining bounds of integrals of the form: $$\int_{\mathbb{R}} |P^{(k)}(t,z-z_0)|dz\leq \mbox{some function of t and }z_0$$ where $P(t,z)$ the density of a normal distribution with mean $0$ and variance $t$, i.e. $P(t,z)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{z^2}{2t}}$ and $(k)$ denotes the $k$-th derivative with respect to $z$.
I am also interested in bounding integrals of the form: $$\int_{\mathbb{R}} |z|^p |z_0|^q|D^{\alpha}P(t,z-z_0)|dz\leq \mbox{some function of t and }z_0$$ for integers $p,q\geq 0$ but I have the impression that this is a particular case of the one above, since the derivative of $P$ are polynomials times $P$.
In general, is there any book or article giving bound for integrals of derivatives of the non-centered Gaussian distribution?
Thank you very much!!