  Local and global weighted norm estimates involving Muckenhoupt weights are
obtained for gradient of solutions to linear elliptic Dirichlet boundary value
problems in divergence form over a Lipschitz domain $\Omega$. The gradient
estimates are obtained in weighted Lebesgue and Lorentz spaces, which also
yield estimates in Lorentz-Morrey spaces as well as H\"older continuity of
solutions. The significance of the work lies on its applicability to very weak
solutions (that belong to $W^{1,p}_{0}(\Omega)$ for some $p>1$ but not
necessarily in $W^{1,2}_{0}(\Omega)$) to inhomogeneous equations with
coefficients that may have discontinuities but have a small mean oscillation.
The domain is assumed to have a Lipschitz boundary with small Lipschitz
constant and as such allows corners. The approach implemented makes use of
localized sharp maximal function estimates as well as known regularity
estimates for very weak solutions to the associated homogeneous equations. The
estimates are optimal in the sense that they coincide with classical weighted
gradient estimates in the event the coefficients are continuous and the domain
has smooth boundary.
