Multiplier-formulation weighted gradient estimate

Establish the weighted gradient estimate for the multiplier formulation of the matrix-weighted elliptic equation, with auxiliary weight \(\mu=|\mathbb{M}|^{\gamma-p}\) and integrals taken with respect to \(\mu\omega\,dx\), under the assumptions \(\omega\in A_p\) and \(u\in W^{1,p-\delta_0}_{\omega^{1-\delta_0/p}}(\Omega)\) is a very weak solution, namely the estimate comparing the \(\gamma\)-integrability of \(\mathbb{M}Du\) on \(B_R\) with the \((p-\delta_0)\)-integrability of \(\mathbb{M}Du\) and the \(\gamma\)-integrability of \(\mathbb{M}\mathbf{f}\) on \(B_{2R}\).

Background

The paper proves near-natural-exponent Calderón–Zygmund estimates for very weak solutions of matrix-weighted quasilinear elliptic equations in a measure-theoretic formulation, using the scalar weight $\omega=|\mathbb{M}|^p$. The authors note that an alternative multiplier formulation can also be treated using the auxiliary weight $\mu=|\mathbb{M}|^{\gamma-p}$, so that the relevant measure becomes $\mu\omega\,dx$.

The proposed estimate would control the weighted $L^\gamma$-norm of the intrinsic gradient $\mathbb{M}Du$ on a smaller ball by a power of its lower-integrability norm on a larger ball together with the corresponding $L^\gamma$-norm of $\mathbb{M}\mathbf{f}$. The authors explicitly state that they expect this estimate to hold but do not establish it in the paper, leaving it as an unresolved extension of their results.