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 $\left(x,t\right)\in {\mathbb{R}}^{N+1},N\ge q\ge 1$. 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% $$\{\begin{array}{ll}\mathcal{L}u=f& \text{in ΩT}\\ u(\cdot ,0)=g& \text{in RN}\end{array}$$ is proved, for $f\in L^p({\Omega }_T),g\in W_X^{2,p}({\mathbb{R}}^N)$.