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

Abstract: 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.