Schauder estimates for Kolmogorov-Fokker-Planck operators with coefficients measurable in time and H\"{o}lder continuous in space

Abstract

We consider degenerate Kolmogorov-Fokker-Planck operators $$ \mathcal{L}u=\sum_{i,j=1}^{q}a_{ij}(x,t)\partial_{x_{i}x_{j}}^{2}u+\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}u-\partial_{t}u,\qquad (x,t)\in\mathbb{R}^{N+1},N\geq q\geq1 $$ 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 H\"{o}lder continuous in space (w.r.t. some distance induced by $\mathcal{L}$ in $\mathbb{R}^{N}$) and only bounded measurable in time; the matrix $\{ a_{ij}\}_{i,j=1}^{q}$ is symmetric and uniformly positive on $\mathbb{R}^{q}$. We prove "partial Schauder a priori estimates" the kind $$ \sum_{i,j=1}^{q}\Vert\partial_{x_{i}x_{j}}^{2}u\Vert_{C_{x}^{\alpha}(S_{T})}+\Vert Yu\Vert_{C_{x}^{\alpha}(S_{T})}\leq c\left\{ \Vert\mathcal{L}u\Vert _{C_{x}^{\alpha}(S_{T})}+\Vert u\Vert_{C^{0}(S_{T})}\right\} $$ for suitable functions $u$, where $$ \Vert f\Vert_{C_{x}^{\alpha}(S_{T})}=\sup_{t\leq T}\sup_{x_{1},x_{2}\in\mathbb{R}^{N},x_{1}\neq x_{2}}\frac{\left\vert f\left( x_{1},t\right) -f\left( x_{2},t\right) \right\vert }{\left\Vert x_{1}-x_{2}\right\Vert ^{\alpha}}. $$ We also prove that the derivatives $\partial_{x_{i}x_{j}}^{2}u$ are locally H\"{o}lder continuous in space and time while $\partial_{x_{i}}u$ and $u$ are globally H\"{o}lder continuous in space and time.