**Notations**. Let $(X, \mathcal{B})$ be a separable Banach space, with its Borel sigma-algebra, $\|\cdot\|$ stands for the norm in $X$, $\mathcal{P}(X)$ - the set of all probability measures on $X$. Let $P(x,B)$ be a stochastic kernel with source and target spaces both equal to $(X, \mathcal{B})$. Let also

$$ L_b(X)=\left\{f\in C_b(X): \mathrm{Lip}(f):=\sup_{x_1,x_2\in X}\frac{|f(x_1)-f(x_2)|}{\|x_1-x_2\|}<\infty \right\},\\ \|f\|_L:=\|f\|_{C(X)}+\mathrm{Lip}(f). $$

By $\|\cdot\|_{TV}$, $\|\cdot\|^*_L$ we denote total variation and dual Lipschitz distances on $\mathcal{P}(X)$: $$ \|\mu_1-\mu_2\|_{TV}=\frac{1}{2}\sup_{f\in C_b(X),\\ \|f\|_{C(X)}\leq 1}\left|(f,\mu_1)-(f,\mu_2)\right|,\ \mu_1,\mu_2\in \mathcal{P}(X);\\ \|\mu_1-\mu_2\|^{*}_{L}=\sup_{f\in C_b(X),\|f\|_{L}\leq 1}|(f,\mu_1)-(f,\mu_2)|,\ \mu_1,\mu_2\in \mathcal{P}(X); $$ By $\|\cdot\|_K$ we denote Kantorovich distance on $$\mathcal{P}_1(X)=\{\mu\in\mathcal P(X):\int_X\|x-x_0\|\mu(dx)<\infty\mbox{ for some } x_0\in X\}:$$ $$ \|\mu_1-\mu_2\|_{K}=\sup_{f\in L_b(X),\\\mathrm{Lip}(f)\leq 1}|(f,\mu_1)-(f,\mu_2)|,\ \mu_1,\mu_2\in \mathcal{P_1}(X). $$

**Question**. If for all $x_1,x_2\in X$ we have $\|P(x_1,\cdot)-P(x_2,\cdot)\|_{TV}\leq \gamma$ for some $\gamma\in (0,1)$, then one can show that the map
$$
G:\mathcal P(X)\to \mathcal P(X),\ G(\mu)(B)=\int_XP(x,B)\mu(dx)
$$
is contraction
$$
\|G(\mu_1)-G(\mu_2)\|_{TV}\leq \gamma \|\mu_1-\mu_2\|_{TV}.
$$
Is it possible to obtain similar contraction result

1. in dual Lipschitz distance, if for all $x_1,x_2\in X$ $\|P(x_1,\cdot)-P(x_2,\cdot)\|^*_{L}\leq \gamma$ for some $\gamma\in (0,1)$?

2. in Kantorovich distance, if for all $x_1,x_2\in X$ $\|P(x_1,\cdot)-P(x_2,\cdot)\|^*_{K}\leq \gamma$ for some $\gamma\in (0,1)$?

If yes, please, could you give an idea of the proof or a reference?