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Register a new user$W^{1,p}$ regularity of solutions to Kolmogorov equation and associated Feller semigroup
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In $mathbb R^d$, $d geq 3$, consider the divergence and the non-divergence form operators begin{equation} tag{$i$} - abla cdot a cdot abla + b cdot abla, end{equation} begin{equation} tag{$ii$} - a cdot abla^2 + b cdot abla, end{equation} where $a=I+c mathsf{f} otimes mathsf{f}$, the vector fields $ abla_i mathsf{f}$ ($i=1,2,dots,d$) and $b$ are form-bounded (this includes the sub-critical class $[L^d + L^infty]^d$ as well as vector fields having critical-order singularities). We characterize quantitative dependence on $c$ and the values of the form-bounds 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. The latter allows to run an iteration procedure $L^p rightarrow L^infty$ that yields associated with ($i$), ($ii$) $L^q$-strong Feller semigroups.
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