  Let $n\ge2$ and $\Omega\subset\mathbb{R}^n$ be a bounded NTA domain. In this
article, the authors investigate (weighted) global gradient estimates for
Dirichlet boundary value problems of second order elliptic equations of
divergence form with an elliptic symmetric part and a BMO anti-symmetric part
in $\Omega$. More precisely, for any given $p\in(2,\infty)$, the authors prove
that a weak reverse H\"older inequality with exponent $p$ implies the global
$W^{1,p}$ estimate and the global weighted $W^{1,q}$ estimate, with $q\in[2,p]$
and some Muckenhoupt weights, of solutions to Dirichlet boundary value
problems. As applications, the authors establish some global gradient estimates
for solutions to Dirichlet boundary value problems of second order elliptic
equations of divergence form with small $\mathrm{BMO}$ symmetric part and small
$\mathrm{BMO}$ anti-symmetric part, respectively, on bounded Lipschitz domains,
quasi-convex domains, Reifenberg flat domains, $C^1$ domains, or (semi-)convex
domains, in weighted Lebesgue spaces. Furthermore, as further applications, the
authors obtain the global gradient estimate, respectively, in (weighted)
Lorentz spaces, (Lorentz--)Morrey spaces, (Musielak--)Orlicz spaces, and
variable Lebesgue spaces. Even on global gradient estimates in Lebesgue spaces,
the results obtained in this article improve the known results via weakening
the assumption on the coefficient matrix.
