Global Sobolev theory for Kolmogorov-Fokker-Planck operators with coefficients measurable in time and $VMO$ in space

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 $\left( x,t\right) \in\mathbb{R}^{N+1},N\geq q\geq1$. We assume that $a_{ij}\in L^{\infty}\left( \mathbb{R}^{N+1}\right) $, the matrix $\left\{ a_{ij}\right\} $ is symmetric and uniformly positive on $\mathbb{R}^{q}$, and the drift \[ Y=\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}-\partial_{t} \] has a structure which makes the model operator with constant $a_{ij}$ hypoelliptic, translation invariant w.r.t. a suitable Lie group operation, and $2$-homogeneus w.r.t. a suitable family of dilations. We also assume that the coefficients $a_{ij}$ are $VMO$ w.r.t. the space variable, and only bounded measurable in $t$. We prove, for every $p\in\left( 1,\infty\right) $, global Sobolev estimates of the kind: \begin{align*} \Vert u\Vert _{W_{X}^{2,p}(S_{T})} \equiv & \sum_{i,j=1}^{q}\Vert u_{x_{i}x_{j}}\Vert_{L^{p}(S_{T})} +\Vert Yu\Vert _{L^{p}(S_{T})} +\sum_{i=1}^{q}\Vert u_{x_{i}}\Vert _{L^{p}(S_{T})} +\Vert u\Vert _{L^{p}(S_{T})} \\ & \leq c\big\{ \Vert \mathcal{L}u\Vert _{L^{p}(S_{T})}+\Vert u\Vert_{L^{p}(S_{T})}\big\} \end{align*} with $S_{T}=\mathbb{R}^{N}\times\left( -\infty,T\right) $ for any $T\in(-\infty,+\infty]$. Also, the well-posedness in $W_{X}^{2,p}(\Omega_{T})$, with $\Omega_{T}=\mathbb{R}^{N}\times(0,T) $ and $T\in\mathbb{R}$, of the Cauchy problem% $$ \begin{cases} \mathcal{L}u=f & \text{in $\Omega_{T}$} \\ u(\cdot,0) =g & \text{in $\mathbb{R}^{N}$} \end{cases} $$ is proved, for $f\in L^{p}(\Omega_{T}), g\in W_{X}^{2,p}(\mathbb{R}^{N})$.