Then

-   for every $(x,\Psi)\in {\mathbb{R}^{d}}\times{\mathcal P}_1(Y)$, there exists $M_{\mathcal T}>0$ such that the operator norm satisfies $$\label{T120}
    \lVert {\mathcal T}^*(x,\Psi)\rVert_{{\mathcal L}({\mathcal F}(U);{\mathcal F}(U))}{\leqslant}M_{\mathcal T}(1+\lvert x\rvert+m_1(\Psi));$$

-   for every $R>0$ there exists $L_{{\mathcal T},R}>0$ such that, for every $(x^1,\Psi^1),(x^2,\Psi^2)\in B_R\times{\mathcal P}(B_R^Y)$, $$\label{T009}
    \lVert {\mathcal T}^*(x^1,\Psi^1)-{\mathcal T}^*(x^2,\Psi^2)\rVert_{{\mathcal L}({\mathcal F}(U);{\mathcal F}(U))}{\leqslant}L_{{\mathcal T},R}\big(|x^1-x^2|+{\mathcal W}_1(\Psi^1,\Psi^2)\big),$$

-   for every $R>0$ there exists $\delta_R>0$ such that $$\label{T119}
    {\mathcal T}^*(x,\Psi)+\delta_R I{\geqslant}0,$$

To see that (T\*2) holds, we apply the definition of operator norm $$\label{T107}
\lVert {\mathcal T}^*(x,\Psi)\rVert_{{\mathcal L}({\mathcal F}(U);{\mathcal F}(U))}\coloneqq \sup\big\{ \lVert {\mathcal T}^*(x,\Psi)\mu\rVert_{{\mathrm{BL}}}: \mu\in{\mathcal F}(U),\lVert\mu\rVert_{{\mathrm{BL}}}{\leqslant}1\big\}.$$ For $(x^1,\Psi^1),(x^2,\Psi^2)\in B_R\times{\mathcal P}(B_R^Y)$, we have, $$\begin{split}
\lVert {\mathcal T}^* & (x^1,\Psi^1)- {\mathcal T}^*(x^2,\Psi^2)\rVert_{{\mathcal L}({\mathcal F}(U);{\mathcal F}(U))}\\
=& \sup\big\{ \lVert ({\mathcal T}^*(x^1,\Psi^1)-{\mathcal T}^*(x^2,\Psi^2))\mu\rVert_{{\mathrm{BL}}}: \mu\in{\mathcal F}(U),\lVert\mu\rVert_{{\mathrm{BL}}}{\leqslant}1\big\} \\
=& \sup\Big\{ \sup\big\{ \langle ({\mathcal T}^*(x^1,\Psi^1)-{\mathcal T}^*(x^2,\Psi^2))\mu, \varphi\rangle: \lVert\varphi\rVert_{{\operatorname{Lip}}}{\leqslant}1\big\} : \mu\in{\mathcal F}(U),\lVert\mu\rVert_{{\mathrm{BL}}}{\leqslant}1\Big\} \\
=& \sup\Big\{ \sup\big\{ \langle \mu, ({\mathcal T}(x^1,\Psi^1)-{\mathcal T}(x^2,\Psi^2))\varphi\rangle:  \lVert\varphi\rVert_{{\operatorname{Lip}}}{\leqslant}1\big\} : \mu\in{\mathcal F}(U),\lVert\mu\rVert_{{\mathrm{BL}}}{\leqslant}1\Big\} \\
{\leqslant}& \sup\Big\{ \sup\big\{ \lVert\mu\rVert_{{\mathrm{BL}}} \lVert({\mathcal T}(x^1,\Psi^1)-{\mathcal T}(x^2,\Psi^2))\varphi\rVert_{{\operatorname{Lip}}}:  \lVert\varphi\rVert_{{\operatorname{Lip}}}{\leqslant}1\big\} : \mu\in{\mathcal F}(U),\lVert\mu\rVert_{{\mathrm{BL}}}{\leqslant}1\Big\} \\
{\leqslant}& \sup\big\{ \lVert({\mathcal T}(x^1,\Psi^1)-{\mathcal T}(x^2,\Psi^2))\varphi\rVert_{{\operatorname{Lip}}}:  \lVert\varphi\rVert_{{\operatorname{Lip}}}{\leqslant}1\big\} \\
{\leqslant}& \lVert {\mathcal T}(x^1,\Psi^1)-{\mathcal T}(x^2,\Psi^2)\rVert_{{\mathcal L}({\operatorname{Lip}}(U);{\operatorname{Lip}}(U))}{\leqslant}L_{{\mathcal T},R}\big(|x^1-x^2|+{\mathcal W}_1(\Psi^1,\Psi^2)\big),
\end{split}$$ and (T\*2) follows from (T2). A similar argument using also gives (T\*1). To prove (T\*3), ler $R>0$ be fixed and let $\delta_R>0$ be such that holds. Then $${\mathcal T}^*(x,\Psi)+\delta_R I=({\mathcal T}(x,\Psi)+\delta_R 