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.
Categories: math.AP math.PR