Global BMO-Sobolev Estimates for Second-Order Linear Elliptic Equations on Lipschitz Domains

Let $n\ge 2$ and $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a bounded Lipschitz domain. In this article, we establish first-order global regularity estimates in the scale of BMO spaces on $\mathrm{\Omega }$ for weak solutions to the second-order elliptic equation $\mathrm{d}\mathrm{i}\mathrm{v}(A\mathrm{\nabla }u)=\mathrm{d}\mathrm{i}\mathrm{v},\mathit{f}$ in $\mathrm{\Omega }$. This is achieved under minimal regularity assumptions on $\mathrm{\Omega }$ and the coefficient matrix $A$, utilizing the pointwise multiplier characterization of the BMO space on $\mathrm{\Omega }$. As an application, we also obtain global estimates of $\mathrm{\nabla }u$ in the Lebesgue space $L^1(\mathrm{\Omega })$ when $\mathit{f}$ belongs to the Hardy space on $\mathrm{\Omega }$.