Zvonkin's transform and the regularity of solutions to double divergence form elliptic equations

Abstract: We study qualitative properties of solutions to double divergence form elliptic equations (or stationary Kolmogorov equations) on~$\mathbb{R}^d$. It is shown that the Harnack inequality holds for nonnegative solutions if the diffusion matrix $A$ is nondegenerate and satisfies the Dini mean oscillation condition and the drift coefficient $b$ is locally integrable to a power $p>d$. We establish new estimates for the $L^p$-norms of solutions and obtain a generalization of the known theorem of Hasminskii on the existence of a probability solution to the stationary Kolmogorov equation to the case where the matrix $A$ satisfies Dini's condition or belongs to the class VMO. These results are based on a new analytic version of Zvonkin's transform of the drift coefficient.