On Energy Inequalities

We consider the Schr¨odinger operator Hsy D qq in a bounded domain D 2 in Rd, dG3, with q in the Kato class Kd D. and finite gauge g x.s Exwexp H0tq Xs.dsx, where Xt.is a Brownian motion and t is the first exit time of the Brownian motion Xt. from the domain D. Let K and G denote the Green operators in D for H and D , respectively. We prove that there is a positive 2 constant a such that