Sobolev estimates for Kolmogorov-Fokker-Planck operators with coefficients measurable in time and $VMO$ in space

We consider Kolmogorov-Fokker-Planck operators of the form \[ Lu=∑i,j=1qaᵢⱼ(x,t)u_{xᵢxⱼ}+∑k,j=1Nbⱼₖxₖu_{xⱼ}-∂ₜu, \] with ( x,t) N+1,N≥ q≥1. We assume that aᵢⱼ∈ L∞( RN+1), the matrix ᵢⱼ\ is symmetric and uniformly positive on Rq, and the drift% \[ Y=∑k,j=1Nbⱼₖxₖ∂_{xⱼ}-∂ₜ% \] has a structure which makes the model operator with constant aᵢⱼ hypoelliptic, translation invariant w.r.t. a suitable Lie group operation, and $2$-homogeneus w.r.t. a suitable family of dilations. We also assume that the coefficients aᵢⱼ are $VMO$ w.r.t. the space variable, and only bounded measurable in t. We prove, for every p∈( 1,∞), global Sobolev estimates of the kind:% {align*} & ∑i,j=1q u_{xᵢxⱼ} _{Lᵖ( RN+1) }+ Yu _{Lᵖ( RN+1) }+∑ᵢ₌₁q u_{xᵢ} _{Lᵖ( RN+1) }\\ & ≤ c\{ Lu _{Lᵖ( RN+1) }+ u _{Lᵖ( RN+1) }\} . {align*}