$L^$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 :$$\|Δ\_{x}^{\frac α2} v \|\_{L^p({\mathbb R}^N)} \lec\_p\Big \| L\_{x } v + \sum\_{i,j=1}^{N}a\_{ij}z\_{i}\partial\_{z\_{j}} v \Big \|\_{L^p({\mathbb R}^N)}, \;\; 1<p<\infty,$$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 $\Delta \_x^{\frac{\alpha }{2}}$ 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$\Delta \_x+\sum \_{i,j=1}^{N}a\_ijz\_i\partial \_z\_j$ satisfy a weak type Hörmander 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 Hörmander type conditions. We complete our result on $L^p$-regularity for $L\_x+\sum \_{i,j=1}^{N}a\_ijz\_i\partial \_z\_j$ 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 $\alpha \_i\in (0,1\wedge \alpha )$ 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.

References in corpus (4)

• Uniqueness of stable-like processes
• $L^$p Estimates For Degenerate Non-Local Kolmogorov Operators
• $L^p$-maximal hypoelliptic regularity of nonlocal kinetic Fokker-Planck operators
• An $L_q(L_p)$-theory for parabolic pseudo-differential equations: Calderón-Zygmund approach

Cited by in corpus (4)

• Global $L_p$ estimates for kinetic Kolmogorov-Fokker-Planck equations in nondivergence form
• $L^$p Estimates For Degenerate Non-Local Kolmogorov Operators
• About the regularity of degenerate non-local Kolmogorov operators under diffusive perturbations
• Schauder estimates for degenerate Lévy Ornstein-Ulhenbeck operators