On Energy Inequalities

AbstractWe consider the Schrödinger operator H=−Δ2+q in a bounded domain D in Rd, d≥3, with q in the Kato class Kd(D) and finite gauge g(x)=Ex[exp∫τ0q(Xs)ds], where (Xt) is a Brownian motion and τ 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 Δ2, respectively. We prove that there is a positive constant α such that1αGf,f≤Kf,f≤αGf,f,where f is a real, measurable function for which (G|f|,|f|) is finite. As a direct consequence of this double inequality, we have that the potential Gf is in the Sobolev space H10(D) if and only if Kf∈H10(D)