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

We consider degenerate Kolmogorov-Fokker-Planck operators Lu=∑ I, j = 1 q a_ij (x,t) ∂ x_i x_j ^ 2 u+∑ k, j = 1 N b_jk x_k ∂ x_j u − ∂_t u,(x,t)∈R^N+1, N ≥ q ≥ 1 such that the corresponding model operator having constant a_ij is hypoelliptic, translation invariant w.r.t. a Lie group operation in 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 L in R^N) and only bounded measurable in time; the matrix {a_ij} I, j = 1 q is symmetric and uniformly positive on R^q. We prove “partial Schauder a priori estimates” of the kind ∑ I, j = 1 q ‖∂ x_i x_j ^ 2 u‖C_x^α(S_T)+‖Y_u‖C_x^α(S_T) ≤ c {‖Lu‖C_x^α(S_T) + ‖u‖C^0(S_T)} for suitable functions u, where Yu=∑ k, j = 1 N b_jk x_k ∂ x_j u−∂ t u and ‖f‖C_x^α(S_T) = sup(t ≤ T_x1,x2∈R^N), sup(x1≠x2) |f(x_1, t), f(x_2, t)|/||x_1-x_2||^α +‖f‖L^∞(S_T). We also prove that the derivatives ∂ x_i x_j^2 u are locally Hölder continuous in space and time while ∂ x_i u and u are globally Hölder continuous in space and time.