a constant $L_{v,R}>0$ such that $$\label{T100}
    \lvert v_\Psi(y^1)-v_\Psi(y^2)\rvert {\leqslant}L_{v,R}\lVert y^1-y^2 \rVert_{\overline Y}\,;$$

-   for every $R>0$, for every $\Psi\in{\mathcal P}(B_R^Y)$, there exists a constant $L_{v,R}>0$ such that for every $y\in B_R^{Y}$, and for every $\Psi^1,\Psi^2\in{\mathcal P}(B_R^Y)$ $$\label{T101}
    \lvert v_{\Psi^1}(y)-v_{\Psi^2}(y)\rvert {\leqslant}L_{v,R}{\mathcal W}_1(\Psi^1,\Psi^2);$$

-   there exists $M_v>0$ such that for every $y\in Y$ and for every $\Psi\in {\mathcal P}_1(Y)$ there holds $$\label{T102}
    \lvert v_\Psi(y)\rvert {\leqslant}M_v \big(1+\lVert y \rVert_{\overline Y} + m_1(\Psi)\big).$$

We now describe the assumptions on ${\mathcal T}$. For $(x,\Psi)\in{\mathbb{R}^{d}}\times{\mathcal P}_1(Y)$, let ${\mathcal T}(x,\Psi)\colon {\operatorname{Lip}}(U)\to {\operatorname{Lip}}(U)$ be an operator such that

-   for every $(x,\Psi)\in{\mathbb{R}^{d}}\times{\mathcal P}_1(Y)$, constants are in the kernel of ${\mathcal T}(x,\Psi)$, that is $$\label{T122}
    {\mathcal T}(x,\Psi)1=0;$$

-   for every $(x,\Psi)\in {\mathbb{R}^{d}}\times{\mathcal P}_1(Y)$, there exists a constant $M_{\mathcal T}>0$ such that the operator norm satisfies $$\label{T111}
    \lVert {\mathcal T}(x,\Psi)\rVert_{{\mathcal L}({\operatorname{Lip}}(U);{\operatorname{Lip}}(U))}{\leqslant}M_{\mathcal T}\big(1+\lvert x\rvert+m_1(\Psi)\big);$$

-   for every $R>0$ there exists $L_{{\mathcal T},R}>0$ such that, for every $(x^1,\Psi^1),(x^2,\Psi^2)\in B_R\times{\mathcal P}(B_R^Y)$, $$\label{T007}
    \lVert {\mathcal T}(x^1,\Psi^1)-{\mathcal T}(x^2,\Psi^2)\rVert_{{\mathcal L}({\operatorname{Lip}}(U);{\operatorname{Lip}}(U))}{\leqslant}L_{{\mathcal T},R}\big(|x^1-x^2|+{\mathcal W}_1(\Psi^1,\Psi^2)\big);$$

-   for every $R>0$ there exists $\delta_R>0$ such that for every $(x,\Psi)\in B_R\times{\mathcal P}_1(Y)$ we have $$\label{T118}
    {\mathcal T}(x,\Psi)+\delta_R I{\geqslant}0.$$

The following lemma is easily proved.

\[T020\] Let ${\mathcal T}(x,\Psi)\colon {\operatorname{Lip}}(U)\to {\operatorname{Lip}}(U)$ satisfy (T0)-(T3) above and let ${\mathcal T}^*(x,\Psi)\colon{\mathcal F}(U)\to{\mathcal F}(U)$ be its adjoint. 