Spatial regularity for a class of degenerate Kolmogorov equations

We establish spatial a priori estimates for the solution u to class of dilation invariant Kolmogorov equation, where is assumed only have certain amount regularity in diffusion’s directions, i.e. $$x_{1}, \ldots , x_{m_{0}}$$ . The result that also regular with respect remaining directions. approach we propose based on commutators identities and allows us obtain Sobolev exponent does not depend integrability assumption right-hand side. Lastly, provide new proof optimal regularity.