Schauder Estimates for Kolmogorov-Fokker-Planck Operators with Coefficients Measurable in Time and Hölder Continuous in Space

We consider degenerate Kolmogorov-Fokker-Planck operators \(\mathcal {L}u=\sum \limits _{i,j=1}^{q}a_{ij}(x,t)\partial _{x_{i}x_{j}}^{2}u+\sum \limits _{k,j=1}^{N}b_{jk}x_{k}\partial _{x_{j}}u-\partial _{t}u,\ (x,t)\in \mathbb {R}^{N+1},N\geq q\geq 1\) such that the corresponding model operator having constant \(a_{ij}\) is hypoelliptic, translation invariant w.r.t. a Lie group operation in \(\mathbb {R}^{N+1}\) and 2-homogeneous w.r.t. a family of nonisotropic dilations. The coefficients \(a_{ij}\) are bounded and Hölder continuous in space (w.r.t. some homogeneous norm \(||\cdot ||\) induced by \(\mathcal {L}\) in \(\mathbb {R}^{N}\) ) and only bounded measurable in time; the matrix \(\left \{ a_{ij}\right \} _{i,j=1}^{q}\) is symmetric and uniformly positive definite on \(\mathbb {R}^{q} \) . We prove “partial Schauder a priori estimates” of the kind \(\sum \limits _{i,j=1}^{q}\Vert \partial _{x_{i}x_{j}}^{2}u\Vert _{C_{x}^{\alpha }(S_{T})}+\Vert Yu\Vert _{C_{x}^{\alpha }(S_{T})}\leq c\left \{ \Vert \mathcal {L}u\Vert _{C_{x}^{\alpha }(S_{T})}+\Vert u\Vert _{C^{0}(S_{T})}\right \}\) for suitable functions u, where \(\Vert f\Vert _{C_{x}^{\alpha }(S_{T})}=\sup \limits _{t\leq T}\sup \limits _{x_{1},x_{2} \in \mathbb {R}^{N},x_{1}\neq x_{2}}\frac {\left \vert f\left ( x_{1},t\right ) -f\left ( x_{2},t\right ) \right \vert }{\left \Vert x_{1}-x_{2}\right \Vert ^{\alpha }}+\left \Vert f\right \Vert { }_{L^{\infty }(S_{T})}.\) We also prove that the derivatives \(\partial _{x_{i}x_{j}}^{2}u\) are locally Hölder continuous in space and time while \(\partial _{x_{i}}u\) and u are globally Hölder continuous in space and time.