A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with VMO coefficients

In recent years, there has been considerable interest in extending the well-known Calderon–Zygmund estimates for the Laplacian to more general equations, in particular equations with highest order coefficients lying in the Sarason space VMO. In addition, the analogous estimates with Morrey spaces replacing Lebesgue spaces have been considered. These Morrey space estimates have been proved by refining the proofs for the Lp estimates. We shall show that the Morrey space estimates follow from the Lp estimates via an elementary argument which is very similar to that used by Campanato.