Improving $L^2$ estimates to Harnack inequalities

We consider operators of the form $\mathcal{L}=-L-V$, where $L$ is an elliptic operator and $V$ is a singular potential, defined on a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with Dirichlet boundary conditions. We allow the boundary of $\mathrm{\Omega }$ to be made of various pieces of different codimension. We assume that $\mathcal{L}$ has a generalized first eigenfunction of which we know two sided estimates. Under these assumptions we prove optimal Sobolev inequalities for the operator $\mathcal{L}$, 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.