Applications of Zvonkin's transform to stationary Kolmogorov equations

In this note we develop a new analytic version of Zvonkin’s transform of the drift coefficient of a stationary Kolmogorov equation and apply this transform to derive the Harnack inequality for nonnegative solutions in the case where the diffusion matrix is not locally Sobolev. We also obtain a generalization of the known theorem of Hasminskii on existence of a probability solution to the stationary Kolmogorov equation.