KFP operators with coefficients measurable in time and Dini continuous in space.

We consider degenerate Kolmogorov–Fokker–Planck operators (Formula presented.) (with (x,t)∈R^N+1 and 1≤m_0≤N) such that the corresponding model operator having constant aij 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 matrix (aij)i,j=1 m^0 is symmetric and uniformly positive on Rm^0. The coefficients aij are bounded and Dini continuous in space, and only bounded measurable in time. This means that, setting (Formula presented.) we require the finiteness of ‖aij‖D(ST). We bound ω_uxixj,ST, ‖u_xixj‖L∞(ST) (i,j=1,2,..,m0), ω_Yu,ST, ‖Yu‖L∞(ST) in terms of ω_Lu,ST, ‖Lu‖L∞(ST) and ‖u‖L∞ST, getting a control on the uniform continuity in space of u_xixj,Yu if Lu is bounded and Dini-continuous in space. Under the additional assumption that both the coefficients aij and Lu are log-Dini continuous, meaning the finiteness of the quantity (Formula presented.) we prove that u_xixj and Yu are Dini continuous; moreover, in this case, the derivatives u_xixj are locally uniformly continuous in space and time.