Kinetic Poincaré inequality for higher-order Kolmogorov equations

Let $k\geq 1$, $d\geq 1$, and let $u=u(t,x_1,\dots,x_k,v)$ be a nonnegative weak subsolution of

$$\partial_t u+\sum_{j=2}^{k-1}x_j\cdot\nabla_{x_{j-1}}u+v\cdot\nabla_{x_k}u=\Delta_vu.$$

Set

$$\widetilde Q=(-2-\kappa,0]\times B_{R_0}\times B_{R_0(1+\kappa^{-1})}\times\cdots\times B_{R_0(1+\kappa^{-1})^k},$$ $$Q_1^-=(-2-\kappa,-1-\kappa]\times B_1\times\cdots\times B_1,\qquad Q_1=(-1,0]\times B_1\times\cdots\times B_1,$$

and define $\langle u\rangle_{Q_1^-}=|Q_1^-|^{-1}\int_{Q_1^-}u\,\mathrm d(t,x,v)$. Higher-order kinetic Poincaré conjecture. There exists a universal constant $R_0>0$ such that, for every $\varepsilon\in(0,1)$ and $\kappa>0$,

$$\left\|(u-\langle u\rangle_{Q_1^-})_+\right\|_{L^1(Q_1)}\leq C\varepsilon^{-1}\left\|\nabla_vu\right\|_{L^1(\widetilde Q)}+C\varepsilon^{(k+1)d}\left\|u\right\|_{L^1(Q_1)},$$

where $C=C(d,\kappa)$. This would extend the trajectorial proof of the kinetic Poincaré inequality to the higher-order Kolmogorov equation; the general case remains unproved, with the trajectory construction and especially the analogue of property (iii) presenting the main difficulty.