Gradient Estimates Near the Natural Exponent for Very Weak Solutions to $A_p$-Weighted Quasilinear Elliptic Equations

We establish local Calderón--Zygmund estimates near the natural exponent for very weak solutions to matrix-weighted quasilinear equations\[ - \mathrm{div\,} A_{\mathbb{M}}(x,Du) = - \mathrm{div\,} A_{\mathbb{M}}(x,\mathbf{f}), \qquad A_{\mathbb{M}}(x,ξ)=\mathbb{M}(x)A(x,\mathbb{M}(x)ξ), \] where $A$