| Abstract: |
| 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 :
\[\|\Delta_{x}^{\frac \alpha 2} v \|_{L^p(\R^N)} \le c_p \Big \| L_{x } v + \sum_{i,j=1}^{N}a_{ij}z_{i}\partial_{z_{j}} v \Big \|_{L^p(\R^N)},\]
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}^{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*} \label{new}
\|\Delta_{y_i}^{\frac {\alpha_i} {2}} v \|_{L^p(\R^N)} \le
c_p
\Big \| L_{x } v + \sum_{i,j=1}^{N}a_{ij}z_{i}\partial_{z_{j}} v \Big \|_{L^p(\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. |
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