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We fix $\alpha \in (0, 1)$ and $c>0$. Let $f : \bR^d \to \bR$ and $\ell : \bR^d \to \bR_+$ be measurable such that $\ell$ is a probability density function and that
$$
\| f \|_\infty + \|\ell\|_{C^\alpha_b} + \int_{\bR^d} |z| \ell (z) \diff z \le c.
$$

Let $(p_t)_{t>0}$ be the standard [Gaussian heat kernel](https://www.wikiwand.com/en/Heat_kernel) on $\bR^d$. We define
$$
I_t := \int_{\bR^d \times \bR^d} |f(y)-f(z)| (1+|z|) \ell (y) p_t (y-z) \diff y \diff z.
$$

>Is there a constant $c_1 >0$ (depending only on $d,\alpha, c$) such that $I_t \le c_1 t^{\frac{\alpha}{2}}$ for $t>0$?

Other upper bounds are also welcome. Thank you for your elaboration.