Schauder estimates for Kolmogorov-Fokker-Planck operators with coefficients measurable in time and Hölder continuous in space

Abstract

We consider degenerate Kolmogorov-Fokker-Planck operators $$\mathcal{L}u=\sum _{i,j=1}^q{a}_{ij}(x,t){\partial }_{x_i{x}_j}^{2}u+\sum _{k,j=1}^N{b}_{jk}{x}_k{\partial }_{x_j}u-{\partial }_{t}u,(x,t)\in {\mathbb{R}}^{N+1},N\ge q\ge 1$$ 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 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 $$\sum _{i,j=1}^q\Vert {\partial }_{x_i{x}_j}^{2}u{\Vert }_{C_x^{\alpha }(S_T)}+\Vert Yu{\Vert }_{C_x^{\alpha }(S_T)}\le c\left\{\Vert \mathcal{L}u{\Vert }_{C_x^{\alpha }(S_T)}+\Vert u{\Vert }_{C^0(S_T)}\right\}$$ for suitable functions $u$, where $$\Vert f{\Vert }_{C_x^{\alpha }(S_T)}=\underset{t\le T}{\sup }\underset{x_1,x_2\in {\mathbb{R}}^N,x_1\ne x_2}{\sup }\frac{\left|f\left(x_1,t\right)-f\left(x_2,t\right)\right|}{{\Vert x_1-x_2\Vert }^{\alpha }}.$$ 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.