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$$-\mathrm{d}\mathrm{i}\mathrm{v}{A}_{\mathbb{M}}(x,Du)=-\mathrm{d}\mathrm{i}\mathrm{v}{A}_{\mathbb{M}}(x,\mathbf{f}),A_{\mathbb{M}}(x,\xi )=\mathbb{M}(x)A(x,\mathbb{M}(x)\xi ),$$ where $A$ has $p$-growth and strong monotonicity, and $\mathbb{M}$ is a measurable positive-definite matrix field. We assume that $\mathbb{M}$ has bounded condition number and that $\omega :=\mathrm{\mid }\mathbb{M}{\mathrm{\mid }}^p\in A_p$, without imposing uniform upper or lower bounds on $\mathbb{M}$. This extends the near-natural Calderón–Zygmund theory developed by Adimurthi–Phuc \cite{AP15} to the matrix-degenerate setting: There exists ${\delta }_0>0$ such that every very weak solution $u\in W_{\omega ,\mathrm{l}\mathrm{o}\mathrm{c}}^{p-{\delta }_0}$ satisfies $$\mathbf{f}\in L_{\omega ,\mathrm{l}\mathrm{o}\mathrm{c}}^{\gamma }⟹Du\in L_{\omega ,\mathrm{l}\mathrm{o}\mathrm{c}}^{\gamma }$$ for $p-{\delta }_0\le \gamma \le p+{\delta }_0$. The proof requires handling the lack of energy estimates below the natural exponent and the use of Lipschitz truncation in the weighted setting. We achieve this through comparison estimates, higher integrability, and weighted analysis techniques.