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 :$$\|Δ\_{x}^{\frac α2} v \|\_{L^p({\mathbb R}^N)} \lec\_p\Big \| L\_{x } v + \sum\_{i,j=1}^{N}a\_{ij}z\_{i}\partial\_{z\_{j}} v \Big \|\_{L^p({\mathbb R}^N)}, \;\; 1<p<\infty,$$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 $Δ\_{x}^{\frac α2}$ in terms of symbols, $α\in (0,2)$. We require that when $L\_x$ is replaced by the classical ${\mathbb R}^d$-Laplacian $Δ\_{x}$, i.e., in the limit local case $α=2$, the operator$ Δ\_{x} + \sum\_{i,j=1}^{N}a\_{ij}z\_{i}\partial\_{z\_{j}} $ satisfy a weak type Hörmander condition with invariance by suitable dilations. {Such} estimates were only known for $α=2$. This is one of the first results on $L^p $ estimates for degenerate non-local operators under Hörmander 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*} \|Δ\_{y\_i}^{\frac {α\_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 $α\_i \in (0, { {1\wedge α}})$ depends on $α$ and on the numbers of commutators needed to obtain the $y\_i$-direction). The last estimates are new even in the local limit case $α=2$ which is also considered.