Propagation of regularity in $L^p$-spaces for Kolmogorov type hypoelliptic operators

Consider the following Kolmogorov type hypoelliptic operator $$ \mathscr L:=\mbox{$\sum_{j=2}^n$}x_j\cdot\nabla_{x_{j-1}}+\Delta_{x_n}, $$ where $n\geq 2$ and $x=(x_1,\cdots,x_n)\in(\mathbb R^d)^n =\mathbb R^{nd}$. Let $\{\mathcal T_t; t\geq 0\}$ be the semigroup associated with $\mathscr L$. For any $p\in(1,\infty)$, we show that there is a constant $C=C(p,n,d)>0$ such that for any $f(t, x)\in L^p(\mathbb R \times \mathbb R^{nd})=L^p(\mathbb R^{1+nd})$, $$ \left\|\Delta_{x_j}^{\frac{1}{1+2(n-j)}}\int^{\infty}_0\mathcal T_{t }f(t+s, x)dt\right\|_p\leq C\|f\|_p,\ j=1,\cdots, n, $$ where $\|\cdot\|_p$ is the usual $L^p$-norm in $L^p(\mathbb R^{1+nd}; d s\times d x)$. To show this type of estimates, we first study the propagation of regularity in $L^2$-space from variable $x_n$ to $x_1$ for the solution of the transport equation $\partial_t u+\sum_{j=2}^nx_j\cdot\nabla_{x_{j-1}} u=f$.