Estimates on Green functions of second order differential operators with singular coefficients

We investigate the Green functions $G(x;x′)$ of some second order differential operators on $Rd+1$ with singular coefficients depending only on one coordinate $x0$⁠. We express the Green functions by means of the Brownian motion. Applying probabilistic methods we prove that when $x=(0,x)$ and $x′=(0,x′)$ (here $x0=0$⁠) lie on the singular hyperplanes, then $G(0,x;0,x′)$ is more regular than the Green function of operators with regular coefficients.