L^p Estimates For Degenerate Non-Local Kolmogorov Operators

Let $z=(x,y)\in {\mathbb{R}}^d\times {\mathbb{R}}^{N-d}$, with $1 \le d < N$\mathrm{.}Weproveaprioriestimatesofthefollowingtype:$$|\Delta_{x}^{\frac \alpha 2} v |_{L^p({\mathbb R}^N)} \lec_p\Big | L_{x } v + \sum_{i,j=1}^{N}a_{ij}z_{i}\partial_{z_{j}} v \Big |_{L^p({\mathbb R}^N)}, ;; 1<p<\infty,$$for$v \in C_0^{\infty}({\mathbb R}^N)$,where$L_x$isanon-localoperatorcomparablewiththe${\mathbb R}^d $-fractional Laplacian $\mathrm{\Delta }\mathrm{\_}{x}^{\frac{\alpha }{2}}$ in terms of symbols, $\alpha \in (0,2)$\mathrm{.}Werequirethatwhen$L_x$isreplacedbytheclassical${\mathbb R}^d$-Laplacian$\Delta_{x}$,i\mathrm{.}e\mathrm{.},inthelimitlocalcase$\alpha =2$, the operator$\mathrm{\Delta }\mathrm{\_}x+\sum \mathrm{\_}{i,j=1}^{N}a\mathrm{\_}ijz\mathrm{\_}i\mathrm{\partial }\mathrm{\_}z\mathrm{\_}j$ 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 $L\mathrm{\_}x+\sum \mathrm{\_}{i,j=1}^{N}a\mathrm{\_}ijz\mathrm{\_}i\mathrm{\partial }\mathrm{\_}z\mathrm{\_}j$ 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 $y\mathrm{\_}i$ (here $\alpha \mathrm{\_}i\in (0,1\wedge \alpha )$ depends on $\alpha$ and on the numbers of commutators needed to obtain the $y\mathrm{\_}i$-direction). The last estimates are new even in the local limit case $\alpha =2$ which is also considered.