$W^{1,p}$ regularity of solutions to Kolmogorov equation with Gilbarg-Serrin matrix

TL;DR

This paper establishes $W^{1,p}$ regularity results for solutions to certain Kolmogorov equations with Gilbarg-Serrin type matrices, analyzing how the regularity depends on parameters and vector field singularities.

Contribution

It provides a quantitative characterization of the regularity of resolvents for Kolmogorov operators with singular matrix perturbations and form-bounded vector fields.

Findings

Regularity of resolvents depends on parameters c and δ.

Results include sub-critical and critical vector field classes.

Operators generate positivity-preserving $L^ abla$ contraction semigroups.

Abstract

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.