Regularity for Elliptic Equations with Coefficients of Small Mean Oscillation

Abstract

We give a detailed proof of interior $W^{2,p}$ regularity for uniformly elliptic equations in nondivergence form \[ a^{ij}(x)D_{ij}u+b^i(x)D_i u=f. \] The leading matrix is assumed to have sufficiently small mean oscillation on the balls under consideration. The proof uses a modern real-variable argument: a constant-coefficient harmonic replacement yields a sharp-function estimate for the Hessian, and the Fefferman--Stein and Hardy--Littlewood theorems permit the coefficient error to be absorbed. A parameter estimate, obtained by Agmon's auxiliary-variable argument, supplies a consistent resolvent on all $L^p$ spaces and makes the subsequent gain of integrability non-circular. The first-order term is retained throughout and is controlled by the scale-invariant quantity $R^{1-n/q}\|b\|_{L^q(B_R)}$, with $q>\max\{n,p\}$. As a consequence, coefficients in $\VMO_{\rm loc}$ give the usual local $W^{2,p}$ regularity for every finite $p$.