  A non-homogeneous mixed local and nonlocal problem in divergence form is
investigated for the validity of the global Calder\'on-Zygmund estimate for the
weak solution to the Dirichlet problem of a nonlinear elliptic equation. We
establish an optimal Calder\'on-Zygmund theory by finding not only a minimal
regularity requirement on the mixed local and nonlocal operators but also a
lower level of geometric assumption on the boundary of the domain for the
global gradient estimate. More precisely, assuming that the nonlinearity of the
local operator, whose prototype is the classical $(-\Delta_p)^1$-Laplace
operator with $1<p<\infty$, is measurable in one variable and has a small BMO
assumption for the other variables, while the singular kernel associated with
the nonlocal $(-\Delta_q)^s$-Laplace operator with $0<s<1<q<\infty$ is merely
measurable, and that the boundary of the domain is sufficiently flat in
Reifenberg sense, we prove that the gradient of the solution is as integrable
as that of the associated non-homogeneous term in the divergence form.
