$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\DeclareMathOperator*{\supp}{supp}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bD}{\mathbb{D}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cE}{\mathcal{E}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cG}{\mathcal{G}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\sD}{\mathscr{D}}
\newcommand{\sE}{\mathscr{E}}
\newcommand{\sG}{\mathscr{G}}
\newcommand{\sH}{\mathscr{H}}
\newcommand{\sK}{\mathscr{K}}
\newcommand{\sP}{\mathscr{P}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\eps}{\varepsilon}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
\newcommand{\andd}{\quad \text{and} \quad}
\newcommand{\qtext}{\quad\text}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$
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
$$
\| f \|_\infty + \|\ell\|_{\infty} + \sup_{\substack{x, y \in {\bR}^d \\ x \neq y}} \frac{| \ell (x)-\ell (y)|}{|x-y|^\alpha} + \int_{\bR^d} |y| \ell (y) \diff y \le c.
$$

Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e.,
$$
p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \exp \left ( -\frac{|x|^2}{4 t} \right ).
$$

 We define
$$
I_t := \int_{\bR^d \times \bR^d} |f(x)-f(y)| (1+|y|) \ell (x) p_t (x-y) \diff x \diff y.
$$

>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.