Heat kernel bounds and desingularizing weights. (Q1403836)

The authors study the parabolic operator \(\partial_t -\Delta_x +V(t,x)\) in \(\mathbb{R}^1_+\times \mathbb{R}^d\), \(d\geq 1\), with a potential \(V=V^+-V^-\), \(V^\pm \geq 0\) assumed to be from a parabolic Kato class. Let \(Z_V(t,x;s,y)\) be the kernel corresponding to this operator and \(\Gamma (t,x)\) be the heat kernel. The authors prove that there are constants \(c_i^\pm >0\) and \(\omega_i\) such that \[ c_1^- \exp(\omega_1(t-s))) \Gamma (c_2^-(t-s),x-y)\leq Z_V(t,x;y,s)\leq c_1^+ \exp(\omega_2 (t-s)) \Gamma(c_2^+(t-s),x-y). \]