| Abstract: |
| We consider Kolmogorov--Fokker--Planck operators of the form
$$
\mathcal{L}u=\sum_{i,j=1}^{q}a_{ij}(x,t)u_{x_{i}x_{j}}+\sum_{k,j=1}^{N} b_{jk}x_{k}u_{x_{j}}-\partial_{t}u,
$$
with $(x,t)\in\mathbb{R}^{N+1}$, $N\ge q\ge 1$. We assume that $a_{ij}\in L^{\infty}(\mathbb{R}^{N+1})$, the matrix $\{a_{ij}\}$ is symmetric and uniformly positive on $\mathbb{R}^{q}$, and the drift term
$$
Y=\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}-\partial_{t}
$$
has a structure ensuring that the corresponding model operator with constant coefficients $a_{ij}$ is hypoelliptic, invariant with respect to a suitable Lie group operation, and $2$-homogeneous with respect to a suitable family of dilations.
We further assume that the coefficients $a_{ij}$ belong to $VMO$ with respect to the space variable and are merely bounded measurable in $t$. For every $p\in(1,\infty)$, we prove global Sobolev estimates of the form
\begin{align*}
\|u\|_{W_{X}^{2,p}(S_{T})}\equiv & \sum_{i,j=1}^{q}\|u_{x_{i}x_{j}}\|_{L^{p}(S_{T})}
+\|Yu\|_{L^{p}(S_{T})}
+\sum_{i=1}^{q}\|u_{x_{i}}\|_{L^{p}(S_{T})} \
& +\|u\|_{L^{p}(S_{T})}
\le c\left( \|\mathcal{L}u\|_{L^{p}(S_{T})}+ \|u\|_{L^{p}(S_{T})}\right),
\end{align*}
where $S_{T}=\mathbb{R}^{N}\times(-\infty,T)$ for any $T\in(-\infty,+\infty]$. |
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