Green's function for second order parabolic equations with singular lower order coefficients
<p style='text-indent:20px;'>We construct Green's functions for second order parabolic operators of the form <inline-formula><tex-math id="M1">\begin{document}$ Pu = \partial_t u-{\rm div}({\mathbf A} \nabla u+ {\mathbf b}u)+ {\mathbf c} \cdot \nabla u+du $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ (-\infty, \infty) \times \Omega $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is an open connected set in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. It is not necessary that <inline-formula><tex-math id="M5">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> to be bounded and <inline-formula><tex-math id="M6">\begin{document}$ \Omega = \mathbb{R}^n $\end{document}</tex-math></inline-formula> is not excluded. We assume that the leading coefficients <inline-formula><tex-math id="M7">\begin{document}$ \mathbf A $\end{document}</tex-math></inline-formula> are bounded and measurable and the lower order coefficients <inline-formula><tex-math id="M8">\begin{document}$ \boldsymbol{b} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ \boldsymbol{c} $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M10">\begin{document}$ d $\end{document}</tex-math></inline-formula> belong to critical mixed norm Lebesgue spaces and satisfy the conditions <inline-formula><tex-math id="M11">\begin{document}$ d-{\rm div} \boldsymbol{b} \ge 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ {\rm div}(\boldsymbol{b}-\boldsymbol{c}) \ge 0 $\end{document}</tex-math></inline-formula>. We show that the Green's function has the Gaussian bound in the entire <inline-formula><tex-math id="M13">\begin{document}$ (-\infty, \infty) \times \Omega $\end{document}</tex-math></inline-formula>.</p>