$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 $$ \| f \|_\infty + \|\ell\|_{C^\alpha_b} + \int_{\bR^d} |z| \ell (z) \diff z \le c. $$

Let $(p_t)_{t>0}$ be the standard Gaussian 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 $I_t \le c_1 t$ for $t>0$?

Thank you for your elaboration.