On Kolmogorov Fokker Planck operators with linear drift and time dependent measurable coefficients

We prove the well-posedness of a Cauchy problem of the kind: (u pound = f, in D'(RN x (0, +infinity)), u(x, 0) = g(x), for all x is an element of RN, where f is Dini continuous in space and measurable in time and g satisfies suitable regularity properties. The operator pound is the degenerate Kolmogorov -Fokker -Planck operator q � pound = i,j=1 N Eaij(t) partial differential 2xixj + k,j=1 bkjxk partial differential xj - partial differential t where {aij}qij=1 is measurable in time, uniformly positive definite and bounded while {bij}Nij=1 have the block structure: {bij}Nij=1 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝O ... O O B1 ... O O ... ... ... ... O ... B kappa O which makes the operator with constant coefficients hypoelliptic, 2-homogeneous with respect to a family of dilations and traslation invariant with respect to a Lie group.