Improving L2 estimates to Harnack inequalities

We consider operators of the form L = -L-V, where L is an elliptic operator and V is a singular potential, defined on a smooth bounded domain Omega subset of R-n with Dirichlet boundary conditions. We allow the boundary of Omega to be made of various pieces of different codimension. We assume that L has a generalized first eigenfunction of which we know two-sided estimates. Under these assumptions we prove optimal Sobolev inequalities for the operator L, we show that it generates an intrinsic ultracontractive semigroup and finally we derive a parabolic Harnack inequality up to the boundary as well as sharp heat kernel estimates.