The equivalence of certain heat kernel and Green function bounds (Q1107017)

Let \(H\) be the self-adjoint operator associated with the closure of the form: \[ Q(f)=\int_{\Omega}\sum_{ij}a_{ij}(x)\frac{\partial f}{\partial x_ i}\frac{\partial \bar f}{\partial x_ j} dx+\int_{\Omega}W| f|^ 2 dx \] where \(\Omega\) is a bounded region in \(\mathbb R^ N\), \(a(x)\) is a self-adjoint measurable real matrix with \(0<\alpha \leq a(x)\leq \mu <\infty\) and \(W\) is a potential on \(\Omega\) such that \(| W| \leq \epsilon H_ 0+b\epsilon^{-\beta}\) for all \(\epsilon >0\) and some \(\beta <\infty\) with \(H_ 0\) the self adjoint operator associated with the closure of the first form appearing in \(Q(f)\). If \(K(t,x,y,)\) is the kernel of \(e^{-Ht}\), \(E\) the bottom strictly positive eigenvalue of \(H\) and \(G(x,y)=\int^{\infty}_{0}K(t,x,y)dt\) it is proved the equivalence of the following two hypotheses: (H2) there exists \(C_ 1>0\), \(\alpha >0\) so that \(G(x,y)\geq C_ 1d(x)^{\alpha}d(y)^{\alpha}\) (H3) the eigenfunction \(\phi\) that corresponds to \(E\) satisfies \(\phi (x)\geq C_ 2d(x)^{\alpha}.\) For Lipschitz domains (H3) is proved to be equivalent to Harnack's inequality.