Uniform estimates for elliptic systems with critical Sobolev coefficients

The $W^{1,p}$ estimate for uniformly elliptic operators of divergence form with VMO coefficients is classical. In particular, this provides $W^{1,p}$ estimates for elliptic operators with coefficients in $W^{1,n}$. However, this theory does not yield constants in the estimates controlled by the $W^{1,n}$ norm of the coefficients, since the total $W^{1,n}$ norm does not control the rate of decay for the mean oscillation. In this work, we establish uniform estimates for divergence-form elliptic systems whose coefficients lie in the critical Sobolev–Lorentz spaces $W^{1,(n,s)}$ with $1\le s<\mathrm{\infty }$, have an $L^{\mathrm{\infty }}$-bounded spatially symmetric part, and satisfy an integral ellipticity condition. The constant in the estimate is controlled by the corresponding norms of coefficients and the ellipticity constant.