Lp estimates for degenerate non-local Kolmogorov operators

Let $z = (x,y) \in \R^d \times \R^N-d$, with $1 \le d < N$. We prove a priori estimates of the following type :
\[
\|\Delta_x^\frac \alpha 2 v \|_L^p(\R^N) \le
c_p
\Big \| L_x v + \sum_i,j=1^Na_ijz_i\partial_z_j v \Big \|_L^p(\R^N),
\;\; 1\]
for $v \in C_0^\infty(\R^N)$,
where $L_x$ is a non-local operator comparable with the $\R^d $-fractional Laplacian $\Delta_x^\frac \alpha 2$ in terms of symbols, $\alpha \in (0,2)$.
We require that when $L_x$ is replaced by the classical $\R^d$-Laplacian $\Delta_x$, i.e., in the limit local case $\alpha =2$, the operator
$ \Delta_x + \sum_i,j=1^Na_ijz_i\partial_z_j $ satisfy
a weak type H\"ormander condition with invariance by suitable dilations. Such estimates were only known
for $\alpha =2$.
This is
one of the first results on $L^p $ estimates for degenerate non-local operators under H\"ormander type conditions.
We complete our result on $L^p$-regularity for $ L_x + \sum_i,j=1^Na_ijz_i\partial_z_j $ by proving
estimates like
\beginequation* \labelnew
\|\Delta_y_i^\frac \alpha_i 2 v \|_L^p(\R^N) \le
c_p
\Big \| L_x v + \sum_i,j=1^Na_ijz_i\partial_z_j v \Big \|_L^p(\R^N),
\endequation*
involving fractional Laplacians in the degenerate directions $y_i$ (here $\alpha_i \in (0, 1\wedge \alpha)$ depends on $\alpha $ and on the numbers of commutators needed to obtain the $y_i$-direction). The last estimates are new even in the local limit case $\alpha =2$ which is also considered.