  We establish spatial a priori estimates for the solution u to a class of
dilation invariant Kolmogorov equation, where u is assumed to only have a
certain amount of regularity in the diffusion's directions. The result is that
u is also regular with respect to the remaining directions. The approach we
propose is based on the commutators identities and allows us to obtain a
Sobolev exponent that does not depend on the integrability assumption of the
right-hand side. Lastly, we provide an alternative proof to that of Theorem 1.5
of [9] for the optimal spatial regularity.
