  In $\mathbb R^d$, $d \geq 3$, consider the divergence and the non-divergence
form operators \begin{equation} \tag{$i$} -\Delta - \nabla \cdot (a-I) \cdot
\nabla + b \cdot \nabla, \end{equation} \begin{equation} \tag{$ii$} - \Delta -
(a-I) \cdot \nabla^2 + b \cdot \nabla, \end{equation} where the second order
perturbations are given by the matrix $$a-I=c|x|^{-2}x \otimes x, \quad c>-1.$$
The vector field $b:\mathbb R^d \rightarrow \mathbb R^d$ is form-bounded with
the form-bound $\delta>0$ (this includes a sub-critical class $[L^d +
L^\infty]^d$, as well as vector fields having critical-order singularities). We
characterize quantitative dependence on $c$ and $\delta$ of the $L^q
\rightarrow W^{1,qd/(d-2)}$ regularity of the resolvents of the operator
realizations of ($i$), ($ii$) in $L^q$, $q \geq 2 \vee ( d-2)$ as (minus)
generators of positivity preserving $L^\infty$ contraction $C_0$ semigroups.
