$L^$p Estimates For Degenerate Non-Local Kolmogorov Operators

Let $z = (x,y) \in {\mathbb R}^d \times {\mathbb R}^{N-d}$, with $1 \le d < N$. We prove a priori estimates of the following type :$$\|\Delta\_{x}^{\frac \alpha 2} v \|\_{L^p({\mathbb R}^N)} \lec\_p\Big \| L\_{x } v + \sum\_{i,j=1}^{N}a\_{ij}z\_{i}\partial\_{z\_{j}} v \Big \|\_{L^p({\mathbb R}^N)}, \;\; 1<p<\infty,$$for $v \in C\_0^{\infty}({\mathbb R}^N)$,where $L\_x$ is a non-local operator comparable with the ${\mathbb R}^d$-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 ${\mathbb R}^d$-Laplacian $\Delta\_{x}$, i.e., in the limit local case $\alpha =2$, the operator$\Delta\_{x} + \sum\_{i,j=1}^{N}a\_{ij}z\_{i}\partial\_{z\_{j}}$ 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 $L\_{x } + \sum\_{i,j=1}^{N}a\_{ij}z\_{i}\partial\_{z\_{j}}$ by proving estimates like\begin{equation*} \|\Delta\_{y\_i}^{\frac {\alpha\_i} {2}} v \|\_{L^p({\mathbb R}^N)} \lec\_p \Big \| L\_{x } v + \sum\_{i,j=1}^{N}a\_{ij}z\_{i}\partial\_{z\_{j}} v \Big \|\_{L^p({\mathbb R}^N)},\end{equation*}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.